3068
J. Phys. Chem. 1980, 84, 3068-3072
Theory of Two-Channel Thermal Unimolecular Reactions. 1. General Formulation Th. Just Institut fur Physikallsche Chemie der Verbrennung der DFVLR, 0-7000 Stuttgart, West Germany
and J. Troe* Institut fur Physikallsche Chemle der Universitiit, 0 3 4 0 0 Gottingen, West Germany (Recelved: April 30, 1980; In Final Form: July 8, 1980)
Two-channelthermal unimolecular reactions are analyzed theoretically. Analytical expressions for the branching ratio into the two channels are derived from solutions of the weak collision master equation. The yield of the upper channel sensitively depends on the average energies transferred per collision. The effect of rotations is discussed which can lead to a channel switching.
Introduction The competition between several reaction channels is a common phenomenon in unimolecular reactions with nonthermal activation processes, such as chemical, photochemical, or ionizing excitation.lP2 In these cases highly excited species can be formed, which have energies in excess of the threshold energies of several reaction channels, and which react with simultaneous formation of various reaction products. Such a situation is much less common in thermal unimolecular reactions. Here, the populations of excited states decrease strongly with increasing energy such that in general only the lowest reaction channel is observed. However, an increasing number of systems have been studied in which the threshold energies of at least two channels are very close to each other, and for which the reaction indeed proceeds simultaneously on two different reaction path^.^-^ Among these are the reactions C3H7I + M HI + C3H6 + M -.+ I + C3H7 + M 4
CzH4 + M +
Hz + CzH2 + M H + CzH3 + M
HZCO + M ’ HZ
+
CO + M H + HCO + M c - C ~ H ~+DM H2C=CHCH2D + M ---* HzC=CDCH3 + M +
-+
Such systems are worthwhile to inspect in detail, since they allow one to study molecular properties which are normally not accessible in thermal unimolecular reactions: (i) Under high-pressure conditions, from the branching ratios, one can obtain thermally averaged ratios of the specific rate constants k(E) for the respective competing channels; (ii) under low-pressure conditions, the study of various bath gases via the weak collision efficiencies Pc leads t~ the average energies (AE)transferred per collision. In addition, however, in principle one can gain also some information on the functional form of the collisional transition probability P(E’/E). If it were possible to vary systematically the difference between the reaction thresholds of the competing channels, e.g., by varying isotopic substitution, one might sample the shape of P(E’/@ with some resolution, since the yield of the upper channel depends very sensitively on the probability to cross the energy gap by collision. It should be noted that studies of the branching ratio provide a more direct access to the transition probability than the common analysis of the collision efficiencies ,&. 0022-3654/80/2084-3068$0 1.OO/O
There have been several theoretical calculations of two-channel thermal unimolecular reaction^.^^^-' Fairly early it was recognized by Chow and Wilson6 that, at low pressures, weak collision effects have a marked influence on the branching ratio. Waage and Rabinovitch7 have followed these suggestions and first applied “competitive collisional activation spectroscopy” to the competitive thermal isomerization of labeled cyclopropane, (For a review of this work see ref 7.) A further continuation of this work has been described in ref 5. The analysis of the experiments has been given in terms of numerical solutions of the master equation. Similar to numerical solutions for the one-channel system, one may expect that an analytical solution of the master equation, even if it is limited to simplified models, provides a more transparent picture of how weak collision effects operate in these reactions. We try to derive such solutions in the following. There is another complication which has been overlooked so far. Since the branching ratios sensitively depend on the difference of the threshold energies of the channels, centrifugal effects may influence the effective energy gap. One can imagine reactions where the ordering of the threshold energies is different for rotationally cold and hot molecules such that the energy gap as a function of the total angular momentum switches its sign. Since our understanding of rovibrational weak collision effects even in one-channel thermal unimolecular reactions is still limited,7-13the treatment of rovibrational effects for the two-channel problem is even more difficult. In the following we present transparent analytical expressions for purely vibrational two-channel models first and generalize these to the rovibrational case afterwards. We discuss simple analytical solutions of the master equation for the two-channel problem which give an insight into the relation between weak collision effects and the two-channel branching ratio of the reaction. We consider branching ratios for the high-pressure and low-pressure limits where explicit expressions can be given. The particularly difficult intermediate falloff range, where weak collision effects for the one-channel problem have only recently been treated,l37l4is analyzed in a more qualitative way. An application of the present theory to experimental systems will be given in part 2 of this work.I5
Specific Rate Constants for Competing Unimolecular Processes We consider the two competing unimolecular processes of the species A (eq 1 and 2). PI and Pz denote the A Pi ( 1) +
0 1980 American Chemical Society
The Journal of Physical Chemistty, Vol. 84, No. 23, 1980 3060
Two-Channel Thermal Urlimolecular Reactions
A+Pz
(2)
a)
products of the two channels, respectively, which may be either isomers of A or fragments. The threshold energies of the two channels are denoted by Eoland Eo!, and the specific rate constants for unimolecular reaction at an energy E by kl(E,J) and k,(E,J). According to the statistical theory of unirnolecular processes, the specific rate constants are given by eq 3. Here p(E,J) is the density
ki = wi (4J) / [hp(e,J) I (3) k2(.E,J) = Wz(E,J)/[M&J)I of states of A, and Wl(E,J) in terms of the statistical adiabatic channel mocle113J6*17 indicates the number of open rovibrational reaction channels with the product P1. It is useful to represent the competing unimolecular processes by their specific branching ratios -q
Vz(LS,J) = 1 - V1(E,J) (4) In numerical treatments, of course, the exact expressions for kl(E,J) and k2(E,J) should be used. However, in order to arrive at analytical, but more approximate, solutions in the following, we employ for the purely vibrational twochannel system an exponential model for the specific branching ratio; i.e., we represent eq 4 by Vl(E) = 1 at Eo,IE < Eo,
Flgure 1. Potential curves for two-channel unlmolecular reactions. Full lines are for rotatlonless molecules ( J = 0), dotted lines for rotating molecules (J > 0): (a) both channels with rlgld activated complexes; (b) both channels wlth loose activated complexes; (c) one channel with rigid, one channel wlth loose activated complex.
Vl(l.:) = 1 - V,(E) at Eo,5 E (5) The three parameterie Vl(Eo2), Vl(m), and 6 should be determined by fitting Vl(E) to the real branching ratios for each system. We have calculated specific branching ratios V,(E) explicitly for a number of typical experimental situations. Always eq 5 was a good approximation. Examples for V2(n will be given in part 2 of this work.16 The case of a constant branching ratio is included for 6 -,m. Various examples of two-channel reactions are illustrated in Figure 1by comparison of potential curves for nonrotating and rotating molecules. Figure l a shows two competing elimination reactions, and Figure l b two bond-fission reactions. In Figure IC, for a competing elimination and bond-fission reaction, the ordering of the threshold energies changes with increasing rotation.
with
General Master Equation for a Two-Channel Reaction The rovibrational master equation for a thermal unimolecular reaction of A, which proceeds via the two channels 1and 2, in the framework of ref 11, can be formulated as d[A(E,J)]/dt = -[MI
rJW
* O
d J ’ I m d EZ(E,J) P(E’, 0
lJm
J’/E,J)[A(E,J)] + [MI d J ‘ l m d E ’Z(E’,J? P(E, 0 J/E’,J?[A(E’,J?I .- (kl(E,J) + &E,J)HA(E,J)I (6) Z(E,J) denotes the total energy transfer collision frequencies in collisions between A and the bath gas M; P(E’,J’/E,J) is the corresponding state (E,J+state {E’,J) collisional transition probability; J, is the largest total angular momentum quantum number for which rotationally stable states of A exist. In the formulation of eq 6 a
continuous energy scale has been adopted.
High-pressure Limit of the Unimolecular Reaction At the high-pressure limit [MI m, and at steady state, the relative populations [A(E,,J)I/[AI = h(E,J) KE,J) (7)
-
according to eq 6 approach the equilibrium populations f(E,J). Equation 6 then leads to d[Pll/dt = +[AI d[P,]/dt = +[A]
LJW d J l m d E f(E,J) kl(E,J) 0
r”
*o
d J S - d E f(E,J) k,(E,J) 0
d[A]/dt = -(d[Pi]/dt
+ d[PZ]/dt)
(9)
(10) (11)
We have high-pressure rate constants into channel 1
and into channel 2 kzm=: kT 82 exp( h Q
-$)
where Qi*=
(14) and Q2* follows analogously. The overall branching ratio
3070
The Journal of Physical Chemistry, Vol. 84, No. 23, 1980
Just and Troe
for the high-pressure limit therefore is given by
*
Qi
Q1’
+ Qz’
(16)
In this evaluation, the energy dependence of the vibrational density of states p(E) was approximated by
~ X P
Vibrational Two-Channel Model for the Low-Pressure Limit of the Unimolecular Reaction In the following we consider a purely vibrational model for the solution of the master equation (eq 6) at steady state and at the low-pressure limit [MI 0 of the unimolecular reaction. Since in this limit there is no flow of molecules by collisional transitions from energies E > Eo, to energies E < Eol(we assume Eol< E,,), the solution of the master equation for E < Eolis identical with the case of a one-channelunimolecular reaction. This case has been considered in detail in ref 11. Analytical solutions have been derived for exponential collision models, Le., for collisional transition probabilities of the type
-
P(E’/E) =
-exp(-$) a + r
-1 e
x p (E’- i E)
a+y
in order to have a representation which is internally consistent with the exponential collision model eq 17-19. For integrating eq 25, we insert eq 22 and 27 and obtain
[(
JI (
k1(i2?i2(E))
atE‘I E at E ” E
(17) Detailed balancing, to a first approximation, relates a with Y by 7 E aF,ykT/(. + FEkT) (18) with
In order to evaluate the integral in eq 28, one has to introduce the specific branching ratio Vz(E) = k 2 ( E ) / [ k i ( E )+ (29) Because of the exponential factor in the integral, only a small energy range above Eozcontributes significantly. Therefore, as long as Vz(E)is only a weak function of energy near EO2,the value in the []-bracket is nearly equal to Vz(Eo&If, however, V2(E)changes markedIy with E, either the calculated specific rate constants kl(E) and k2(@ have to be used, or an exponential model of V2(E)like eq 5 has to be employed. For the latter case we get
(It should be emphasized that the exponential collision model cannot be applied when FE is a marked function of the value of Eol.) With this model, for energies E < Eol, the nonequilibrium population is given by eq 20. The
h(E)
1-
FEkT + FEkT
population of the reactive states above the lower threshold energy Eol follows directlyll from the population of the nonreactive states via eq 6, and by the use of detailed balancing P(E/E? f(E? = P(E’/E) f(E), as
Inserting eq 17 and 20 into eq 21 gives for E
(31) kl0 = ko - k2o Equation 30 shows a marked influence of the parameter a on the relative yield kzo/koof the upper channel. For strong collisions, a >> FE k T , eq 30 gives
> Eol For weak collisions, a
K1is always fulfilled. The shapes of the curves obtained are quite similar to model calculations from ref 6, although the fine details, due to the simplicity of this approach, will be different from the results of the exact integration with energy-dependent specific rate constants.
-3
Acknowledgment. J.T. thanks the Deutsche Forschungsgemeinschaft for financial support of his work.
t
References and Notes -2
-1
1 log ko/k,-
0
2
3
Flgure 3. Upper channel yield In the falloff range for strong collisions = (Hinshelwood model, eq 41-45; parameters used: (1) f(E,,)lf(E,,) ’I3,K l / ( K 1 ” K2) = 0.05,K2/(K1’ K2) = 0.08; (2) f(Eo2)/f(E,,) = ’/lo, other parameters as (1); (3) f(Eo2)/f(EO1)= ‘1, other parameters as (1); (4) K*/(K,* K ~=) 0.1, f(E,,)/f(E,,) = K,I(K~* K2) = 0.05; (5)K 2 / ( K l * 4- K2) = 0.5, other parameters as (4)).
+
+
+
+
scribed, e.g., in ref 1,2, and 13. For a first orientation, we apply in the following Hinshelwood‘s assumption k,(E) = constant = K1for Eol I E I EO2,kl(E) = constant = K1* for Eo, IE, and k2(E)= constant = K2for Eo, I E. Then, we have
Sm
f(E) dE (45) K1* + K2 + [MI2 E~ The ratio k2/k is illustrated in Figure 3, for a realistic choice of specific rate constants and equilibrium populations, as a function of ko/k,. k2/k rises monotonically with k2
=
(1) J. Troe in “Physlcal Chemistry. An Advanced Treatise”, Vol. VIb, W. Jost, Ed., Academlc Press, New York, 1975. (2) W. Forst, “Theory of Unlmolecular Reactions", Academic Press, New York, 1973. (3) B. J. Gaynor, R. G. Gilbert, and K. D. King, Chem. Phys. Leff., 58, 591 (1978);K. D.King, D. M. Gdden, 0. N. Spokes, and S.W. Benson, Int. J . Chem. Kinet., 3, 411 (1971). (4) Th. Just, P. Roth, and R. Damm, Symp. (Int.) Combust., [Proc.], 18, 961 (1977); G. Rimpel and Th. Just, Ber. Bunsenges. Phys. Chem., in press. (5) I. E. Klein, B. S.Rablnovltch, and K. H. Jung, J. Chem. Phys., 87, 3833 (1977); I. E. Klein and B. S. Rabinovitch, J. Phys. Chem., 82, 243 (1978). (6) N. Chow and D. J. Wllson, J. Phys. Chem., 68, 342 (1962). (71 . . E. V. Waaae and B. S. Rablnovitch. Chem. Rev.. 70. 377 11970): J. phys. Cbm., 78, 1695 (1972);D: C. Tardy and B. S: Rablriovitch; Chem. Rev., 77, 369 (1977). (8) A. P. Penner and W. Forst, Chem. Phys., 11, 243 (1975); 13, 51 (1976). (9) A. P. Penner, Mol. Phys., 38, 155 (1979). (10) A. J. Stace, Mol. Phys., 38, 155 (1979). (11) J. Troe, J . Chem. Phys., 66, 4745 (1977). (12) J. Troe, J . Chem. Phys., 66, 4758 (1977). (13) J. Troe, J . Phys. Chem., 83, 114 (1979). (14) K. Luther and J. Troe, Synp. (It.)Combust., [Proc.],17,535 (1979). (15) fh. Just and J. Troe, J. Phys. Chem., to be published as part 2 of this work. (16) M. Quack and J. Troe, Ber. Bunsenges. phys. Chem., 78,240 (1974). (17) M. Quack and J. Troe, Gas Kinet. Energy Transfer, 2, 125 (1977). (18) D. C. Tardy and B. S.Rabinovitch, J. Chem. phys., 45,3720 (1966); 48, 1282 (1968). (19) J. E. Dove and S. Raynor, J. Phys. Chem., 83, 127 (1979).
Pressure Dependence of Atom Recombination and Photolytic Cage Effect of Iodine in Solution K. Luther, J. Schroeder, J. Tree,* and U. Unterberg Institut fur Physikalische Chemle, Universitiit Gijttlngen, D-3400 Gijttingen, West Germany (Received: April 10, 1980)
The effect of pressures up to 3 kbar on the photodissociation and subsequent atom recombination of iodine was investigated in n-heptane, isooctane, and methylcyclohexane. It was found that the photodissociation quantum yield 4 decreased markedly with increasing density of the solvent as a consequence of the photolytic cage effect, and a linear correlation between 4 and w-~, where q denotes solvent viscosity, was observed. The atom recombination rate constant showed the expected linear dependence on q-’.
Introduction Chemical reactions in the liquid phase are significantly affected by transport processes in the solvent that control the movement of reactants toward or apart from each other. On the one hand there is the uncorrelated statistical molecular motion leading to large overall displacements on a longer time scale that are described as diffusion of the reactants in a quasicontinuous solvent medium with the appropriate boundary conditions at the reaction surface. From this one can obtain a steady-state, diffusioncontrolled reaction rate constant, which, in its simplest approximation, is inversely proportional to the bulk shear 0022-3~54/a0/2084-3072~0 1.oo/o
viscosity of the so1vent.l The homogeneous recombination of halogen atoms in various solvents has been shown in the past to be well described by this On the other hand there is the correlation of molecular motion in the liquid on a short time scale, including multiple collisions of the same partners in a solvent cage, which may alter reaction product yields as compared to the gas phase. A particularly clear-cut example of this is the reduction of the primsy quantum yield of photodissociation for halogens in solution due to geminate recombination of halogen atoms in the solvent cage, known as the “photolytic cage effect”.lOJ1A systematic study of this effect for the iodine 0 1980 American Chemical Society
