J. Phys. Chem. 1986, 90, 357-365
357
Elementary Reactions in Compressed Gases and Liquids: From Collisional Energy Transfer to Diffuslon Control J. Troe Institut fur Physikalische Chemie der Universitat Gottingen, Gottingen, West Germany (Received: May 30, 1985)
Elementary chemical processes such as unimolecular dissociation and isomerization, atom and radical recombination, and photodissociation have been studied experimentally over wide density ranges, from low-pressure gases via highly compressed gases into compressed liquids. The density dependence reveals various aspects of reactant-solvent interactions: isolated molecule behavior and collisional-energy-transfer control at low gas pressures, control by intramolecular processes in high-pressure gases, the contribution of van der Waals complexes in high-pressure gases, diffusion control and energy relaxation in the liquid phase, solvent shifts in van der Waals complexes and liquid cages. These observations are represented by simple models. Particular emphasis is given to halogen dissociation, recombination,and photolysis and to the photoisomerization of trans-stilbene and diphenylbutadiene.
1. Introduction
Reactions in liquid phase are governed by complicated interactions between reaction partners and solvent molecules. In order to understand these interactions, a study of the reaction under variable density conditions, from the low-pressure gas into the compressed liquid phase, appears useful.’ At very low density there is the possibility of investigating isolated molecule or single binary collision behavior. With increasing density, frequent collisions can populate metastable molecular states to an extent that intramolecular processes become rate determining. At high pressures the formation of van der Waals complexes can modify the molecular properties to some extent. At the highest densities finally diffusion control induced by the viscous character of the surrounding medium sets in. Also, strong solvent-reactant interactions fully develop with a dense phase “solvent shift” of molecular properties. The various reaction domains can be studied experimentally by continuously varying the solvent density from dilute gas to compressed liquid-phase conditions. Unfortunately, this concept so far has been followed only for few experimental systems. On the theoretical side, sufficiently realistic models, describing the reaction over the full density range, are also not available. However, limited ranges can be characterized quite well. This article reviews experiments and theories for a series of model reactions. Recent experimental and theoretical effort has led to a considerable improvement of the situation. It is the aim of this article to discuss mainly the experimental work. Attention will be focused on selected elementary kinetic phenomena such as thermal unimolecular decomposition and the reverse radical recombination reactions, primary photolysis, and unimolecular isomerizations. This group of reactions is characterized by intramolecular rearrangements, intermolecular energy transfer, radiationless electronic transitions in some cases, fragment separation, or reactant approach. There is the transition from free flights between rare collisions in the low-pressure gas to permanent collisional perturbation inducing diffusive motions in dense liquids. There are isolated molecules in the low-pressure gas, simple van der Waals complexes at moderate gas pressures until closed solvation shells are built up, and dense liquid-phase reaction cages around the reactants. Molecular parameters, like frequencies or reaction threshold energies, in general, will be influenced by van der Waals clustering and liquid-phase solvation. Specific reactant-solvent interactions here can be of strongly varying influence. We shall illustrate these various aspects by selected experimental examples. On the theoretical side, at low pressures unimolecular rate theory applies.*-‘ van der Waals complexes already contribute (1) J. Troe in “High Pressure Chemistry”, H. Kelm, Ed., Reidel, Dordrecht, 1978,p 489. (2) P. J. Robinson and K. A. Holbrook, “Unimolecular Reactions”, Wiley-Interscience, London, 1972
0022-3654/86/2090-0357$01.50/0
to atom recombination processes in the low-pressure range.5 They are even more important in compressed gases. The extent of van der Waals clustering in larger complexes and the influence on the kinetics has been demonstrated in many-body trajectory calculations: A large number of studies employed transport equations, like the Fokker-Planck, Smoluchowski, Langevin, and BGK equations, for description of the transition to viscous solution b e h a v i ~ r . ~ . ~The - ’ ~ Kramers-Smoluchowski approachlo here plays a most important role. However, the applicability of continuum models has to be viewed critically. Great effort has been devoted to this aspect recently. Modifications have been introduced which concern the viscoelastic response of the inner and outer solvation shells as well as the frequency dependence of the formal viscosity.7-9v11-13 Besides the transport aspect, solvent shifts in van der Waals clusters and in dense phase cages have been con~idered.’~ Only limited explicit treatments are available for such interactions. Many-body trajectory calculations for the dense liquid provide one possibility for testing phenomenological transport equations of dense phase kinetic^.^,'^-'^ The first steps in this direction have (3) W. Forst, “Theory of Unimolecular Reactions”, Academic Press, New York, 1973. (4) J. Trw in “Physical Chemistry. An Advanced Treatise”, Vol. VIB, W. Jost, Ed., Academic Press, New York, 1975,p 835. (5) D. L. Bunker and N . Davidson, J . Am. Chem. SOC.,80,5090 (1958); K. E. Russell and J . Simons, Proc. R. SOC.London, Ser. A, 217,271 (1953). (6)A . J. Stace and J. N. Murrell, Mol. Phys., 33, 1 (1977);F. G. Amar and B. J. Berne, J . Phys. Chem., 88, 6720 (1984). (7) D.G.Truhlar, W. L.Hase, and J. T. Hynes, J . Phys. Chem., 87,2664 (1983). (8) M. W. Balk, C. L. Brooks, and S. A. Adelman, J . Chem. Phys., 79, 804 (1983);C.L.Brooks, M. W. Balk, and S. A. Adelman, J . Chem. Phys., 79,784 (1983). (9) R. F. Grote and J. T. Hynes, J. Chem. Phys., 73,2715 (1980);74,4465 (1981). (10)H.A. Kramers, Physicu, 7,284(1940);S . Chandrasekhar, Rev. Mod. Phys., 15, l(1943). (11) A.G. Zawadski and J. T. Hynes, Chem. Phys. Lett., 113,476 (1985). (12) B. Carmeli and A . Nitzan, Phys. Rev. A , 29, 1481 (1984);J . Chem. Phys., 79,393 (1983);80,3596 (1984);A. Nitzan, J. Chem. Phys., 82, 1614 ( 1985). (13) M. Borkovec and B. J. Berne, J . Chem. Phys., 82,794(1985);J. E. Straub, M. Borkovec, and B. J. Berne, J . Chem. Phys., 83, 3172 (1985). J. A. Montgomery, D. Chandler, and B. J. Berne, J . Chem. Phys., 70,4056 (1979). (14) J. Schroeder and J. Troe, Chem. Phys. Left., 116,453 (1985). (15) A. H. Lipkus, F. B. Buff, and M. G. Sceats, J . Chem. Phys., 79,4830 (1983); D. L.Bunker and B. S . Jacobson, J. Am. Chem. Soc.,94,1843 (1973); J. N. Murrell, A. J. Stace, and R. Dammel, J . Chem. SOC.,Furuduy Trans. 2,28, 1532 (1978);B.C.Freasier, D. L. Jolly, and S. Nordholm, Chem. Phys., 82,369(1983);R. 0.Rosenberg, B. J. Berne, and D. Chandler, Chem. Phys. Letr., 75, 162 (1980). (16) G.W. Robinson, W. A. Jalenak, and D. Statman, Chem. Phys. Lett., 110,135 (1984). (17) P. Bado, P. H. Berens, J. P. Bergsma, S. B. Wilson, and K. R. Wilson in “Picosecond Phenomena III”, Springer Series CP 23,Springer Berlin, 1982, p 260.
0 1986 American Chemical Society
358
Troe
The Journal of Physical Chemistry, Vol. 90, No. 3, 1986
0
I
a5
1D
I
1.5
,
2.0
[ N ? ] /VI-’ mol cm”
Figure 1. Stern-Volmer plots for collisional quenching of NO, photolysis (@(P)= primary photolysis quantum yield at pressure P; excitation wavelengths X = 404.7 ( O ) , 399 (A),and 366 nm (0). Measurements from ref 23).
,A#
,- m - -o o
3x00
v
/ cm-’
Figure 2. Collisional modification of fluorescence spectra of Si p-difluorobenzene (bottom, unrelaxed spectrum, no 0, added, time between collisions T = 5000 ps; middle, 5900 torr of O2added, T = 39 ps; top, relaxed spectrum, 31200 torr of 0, added, T = 7 ps. Measurements from ref 25).
been made. As described above, reactions from the gas into the liquid phase involve a variety of different “mechanisms” which in ref 22. Nevertheless, yc often can be estimated realistically one may try to separate. In recent transport models general such that the gas pressure provides an “internal clock” or a treatments over the full density range have also been p r ~ p o s e d . ~ * * ~ ~“collisional timing”. Very short times are accessible by applying It remains to be verified, to what extent global descriptions of this high gas pressures; e.g., times between “collisions”are of the order type can account for all details of the specific mechanisms and s at gas pressures of 1000 bar. As an example, primary interactions. In our present review we represent the experiments photolysis quantum yields 4 of NO2 at various wavelengths are by simpler, semiquantitative models which are developed in close shown in Figure 1 as a function of gas pressure.23 The observed relation to experimental observations (see, e.g., ref 20). decrease of 4 at several hundred bar bath gas pressure corresponds to k ( E ) values for photolysisZ4of the order 1012-1013SKI. 2. Experimental Studies over Wide Density Ranges Collisional timing of intramolecular vibrational redistribution Experimental observables such as rate coefficients, transient (IVR) by high gas pressures has been studied by Parmenter and populations of states, or quantum yields are measured at various co-workersZ5in observations of the fine structure of fluorescence gas or liquid pressures. In order to represent these data, various spectra. Figure 2 shows the increasing extent of quenching of possibilities exist. One may, e.g., choose plots of the observables unrelaxed spectra and the growth of relaxed spectra with increasing as a function of the solvent density or concentration [MI. Howpressure, defining time scales for IVR in the picosecond range. ever, a more useful approach consists of replacingZothe solvent So far, direct collision-free measurements of intramolecular density by the inverse of the diffusion coefficient D. D-’ at low processes, such as dissociation, isomerization, and IVR, can only gas pressures increases in proportion to the pressure P or the gas be compared in rare cases with indirect collisional timing exconcentration [MI; in the liquid phase D-I increases in proportion periments on the same system. Nevertheless, there is little doubt to the solvent viscosity 7. In the transition range between gas and that the two techniques can provide similar information. An liquid, D can be derived from the generalized Stokes-Einstein extension of collisional timing experiments into the liquid phase relationship proposed in ref 21 and viscosity measurements. The allows one to investigate the validity of a binary collision concept viscosity 7 in low-pressure gases does not depend on [MI and, in liquids: One may estimate liquid-phase collision frequencies therefore, is not a suitable variable to characterize the effective roughly either by the relationZ6 collision frequency Z . In the liquid phase, small density changes will have a large influence on the kinetics. A plot as a function Z [M]d2/67 (1) of D-I will stretch this representation and will be most adequate (d is the collision diameter) or by scaling up gas-phase Lenfor the proportional dependences of observables on 7 or D-I. nard-Jones collision frequencies ZLJ (number per unit time 2.1. Collisional Timing at High Densities. The most direct throughout this article) viaZ0 observation of intramolecular dynamics, of course, is made with isolated molecules, either cooled in supersonic jets or with thermal energy distributions in bulk gases. If such experiments are not possible, the rates of intramolecular processes can be inferred from where [MIodenotes a low gas density. Equations 1 and 2 are not more indirect studies. Unimolecular reaction rate constants k ( E ) , equivalent and their applicability still has to be validated by e.g., can be obtained from reaction yields when the intramolecular experiments. process occurs in competition with collisional deactivation. Since the deactivation rate coefficient kdeaoin general, will not be equal to the Lennard-Jones collision frequency ZLj, a collision efficiency (22) J. Troe, J. Phys. Chem., 87, 1800 (1983). yc = k,jeac/ZLjhas to be introduced. yc is a complicated function (23) H. Gaedtke and J. Troe, Ber. Bunsenges. Phys. Chem., 79, 184 of energy-transfer and intramolecular quantities such as analyzed (1975). (24) H. Gaedtke, H. Hippler, and J. Troe, Chem. Phys. Lett., 16, 177 (18) M. G. Sceats, J. M. Dawes, and B. P. Millar, Chem. Phys. Lett., 114, 63 (1985); M. G. Sceats, J. M. Dawes, P. M. Rodger, and D.P. Millar, Ber. Bunsenges. Phys. Chem., 89, 233 (1985). (19) M. Borkovec and B. J. Berne, J . Phys. Chem., 89, 3994 (1985). (20) B. Otto. J. Schroeder. and J. Troe. J . Chem. Phvs.. 81. 202 (1984). t21j H. Hippler, V. Schubert, and J. Troe, Ber. Bunsenges. Phys. Chem;, 89, 760 (1985).
( 1972).
(25) R. A. Coveleskie, D. A. Dolson, and C. S. Parmenter, J . Phys. Chem., 89, 645, 655 (1985). (26) D. W. Oxtoby, J . Chem. Phys., 70, 2605 (1979); G. Jager, Sit-
zunesber. Akad. Wiss. Wien. Math. Naturwiss. K1,Abt., 1 1 1 . 697 (1902). 6 7 ) H. Hippler, L. Lindemann, and J. Troe, J . Chem. Phys., 83, 3906 (1985). M. J. Rossi, J. R. Pladziewicz, and J. R. Barker, J . Chem. Phys.. 78, 6695 (1983).
Elementary Reactions in Compressed Gases and Liquids
The Journal of Physical Chemistry, Vol. 90, No. 3, 1986 359
P (He) / k bar
P( He)/bar
krec ,,3 mol. s
Figure 4. Same as Figure 3, representation as a function of [He].
aoi
P (Nel/kbor 01 1 2 4 6
Figure 3. Bromine photolysis quantum yields 4 and pseudo-second-order recombination rate coefficients k,, in the bath gas He (measurements from ref 33; lines, eq 5-9, D = &-He).
Frequent collisions in high-pressure gases can establish nearly equilibrium populations f(E)of excited metastable vibrational energy states. Under these conditions, unimolecular reactions are characterized by rate-determining intramolecular processes in nearly equilibrium ensembles. Falloff curves for the thermal unimolecular reactions28describe the approach of the high-pressure range where the rate constant is given by
k,
L:P(E)f(E) dE
(3)
(Eo is the reaction threshold energy). It should be emphasized that, for basic reasons, eq 3 cannot be realized completely since collisional equilibration of metastable excited states will lead smoothly over to diffusion contr01.~ 2.2. Thermal Decomposition at High Densities. Only limited experimental observations of thermal unimolecular dissociation reactions in dense media are available. Reactions with large dissociation energies would require realizing high temperatures and high pressures at the same time which is technically not easy. Experimental studies on the thermal decomposition of I2 in shock waves a t 1100 K have been to pressures of the bath gas Ar of 300 bar. Under these pressures deviations from second-order rate laws become apparent. Since for diatomic molecules there is nothing like a “high-pressure limit” of the unimolecular dissociation such as characterized by eq 3, the observations have to be attributed to a transition from the collisional-activation-controlled to a diffusion-controlled dissociation reaction. Thermal decomposition reactions in systems with small dissociation energies in compressed liquids have been studied to some extent. The decomposition of di-tert-butyl peroxide has been studied in liquid n-hexane30 up to pressures of 2300 bar in the temperature range 41 3-473 K. Similarly, the decomposition of tert-butyl peroxipivalate in liquid n-hexane was followed up to 2000 bar,31 The pressure dependence of the first-order rate (28) J. Troe, J . Phys. Chem., 83, 114 (1979); Eer. Bunsenges. Phys. Chem., 87, 161 (1983); R. G. Gilbert, K. Luther, and J. Troe,Eer. Bumenges. Phys. Chem., 87, 169 (1983). (29) H. Hippler, K.Luther, and J. Troe, Eer. Bumenges. Phys. Chem., 77, 1104 (1973); J. Troe and H. Gg. Wagner, 2.Phys. Chem. (Frankfurt am Main), 55, 326 (1967). (30) M. Buback and H. Lendle, 2.Naturforsch. A, 34, 1482 (1979). (31) M. Buback and H. Lendle, Z. Narurforsch. A, 36, 1371 (1982).
Figure 5. Same as Figure 3, for the bath gas Ne (D =
coefficient k was represented by the “activation volume” AV defined by a In k AV = -RT(4) aP Whereas a “normal value”32of AV = 10.1 cm3 mol-l was found for the former system, Av* = 1.6 cm3 mol-’ was observed for the latter reaction. Activation volumes AV contain transport cont r i b u t i o n ~as ~ ~well ~ ~as ~ ’solvent ~ shift contributions.’~14The term “activation volume” should, therefore, be understood as a purely formal quantity as defined by eq 4. 2.3. Atom and Radical Recombination Studies over Wide Density Ranges. With laser flash photolysis techniques atom and radical recombination reactions today can be studied relatively easily in high-pressure reaction cells such that wide density ranges can be covered. Such studies in a variety of bath gases have been performed for the recombination of bromine33 and iodine at0ms.20929Figures 3-8 demonstrate a variety of observations. In (32) T. Asano and W. J. Le Noble, Chem. Rev., 78,407 (1978); W. J. Le Noble, Prog. Phys. Org. Chem., 5, 207 (1967). (33) H. Hippler, V. Schubert, and J. Troe, J . Chem. Phys., 81, 3931 (1984).
Troe
360 The Journal of Physical Chemistry, Vol. 90, No. 3, 1986 P ( A d / kbor 1 2 46 1
0.5
0.2 0.1 0.05
0.02
Figure 8. Iodine recombination rate coefficients in gaseous (0)and liquid ethane (e) (D = DI-C2H6; line, eq 5-7, measurements from ref 20).
c
‘VI
7
v
m
8
1 lo2
io3
IS’ / cmm2s
10
Figure 6. Same as Figure 3, for the bath gas Ar (D = kfec in eq 5-9 fitted either to low-pressure (1) or medium-pressure (2) gas-phase
(ii) The phase change gas/liquid in Figure 8 does not introduce discontinuities of the krec(D-’)plot. (iii) For some heavier rare gases, pronounced S-shaped curves were ~ b s e r v e d ~in* ,krw(D-’) ~~ plots at moderate gas pressures where van der Waals clusters must be present. The transition from termolecular atom recombination into the diffusion-controlled range, apart from some “anomalies” with S-shaped curves, follows simple laws. The full lines in Figures 3-8 correspond to the expression
data, D = DBr-Ar).
krec z=
(5)
kdiffis given by the Smoluchowski equation for diffusion-controlled encounters
1
kdiff = ~ R N A R D A - M
0.5
0
k!eckdiff
k?ec -k kdiff
:::
(formulated for identical species, DA-Mis the binary diffusion coefficient of radicals A in solvent M, R is the contact distance). It was shown in ref 20 that
0.05
aoz
R I
1
I
I
1
lo4
1
c
lo” I 10’
(6)
10‘
0-’z x l - 2 s
Figure 7. Same as Figure 3, for the bath gas N2 (D = DBr+; curve a, cluster mechanism of eq 14; curve b, diffusion control mechanism of eq 9; photolysis wavelength X = 628 (0), 590 (0), 530 nm (e)).
the lightest bath gas He, at a pressure close to 6 kbar, a turnover into diffusion control just becomes detectable for bromine atom recombination. Figure 3 shows this in a plot of the pseudo-second-order rate coefficient k,,, as a function of the inverse of the binary diffusion coefficient DBrwH;l.Amazingly, the plot of k,, as a function of [He] in Figure 4 gives an S-shaped curve which does not provide direct evidence for the turnover hidden in the density dependence of D-I. Whereas the transition to diffusion control is difficult to establish for He, heavier bath gases show an earlier turnover. Figures 5-7 give results for Ne, Ar, and N,. Figure 8 demonstrates similar behavior for iodine recombination in gaseous and liquid ethane. A number of points appear worth noticing: (i) The turnover to diffusion control can be realized either in the gas phase or at densities close to the phase transition.
2”*(,A
UM)/~
(7)
provides a satisfactory estimate on the basis of the Lennard-Jones diameters uA and uM. In the framework of simple diffusion theory,34 kfw denotes the hypothetical rate coefficient “in the absence of diffusion control”. The curves in Figures 3-8 have been drawn by identifying kfecwith the low-pressure gas-phase values scaled up, by analogy with eq 2, by assuming kfs D-I. This simple representation of the results works surprisingly well, although minor deviations occur such as the mentioned S-shaped anomalies in some bath gases and a slight “broadening” of the transition compared to eq 5, see e.g. Figure 8. While atom recombination shows a relatively sharp “one-dimensional turnover” from the termolecular range into diffusion control, recombination of polyatomic radicals is characterized by broad high-pressure plateaus corresponding to eq 3. Diffusion control of polyatomic radicals in the liquid phase has been studied systematically and the extent of validity of the Smoluchowski equation has been i n v e ~ t i g a t e d . ~Full ~ transition curves from termolecular lowpressure gas-phase behavior, via a broad intermediate, bimolecular “high-pressure” plateau4 corresponding to eq 3, into the dense fluid diffusion control range have not yet been obtained experimentally. However, experiment^^^ on the recombination of methyl radicals in the bath gas Ar up to pressures of 200 bar show the expected behavior, i.e. a very broad high-pressure plateau with a slight indication of diffusion control decline of the rate coefficient at 200 bar. In contrast to this, a decline of the rate coefficient of (34) S. F. Burlatsky, P. P. Levin, I. V. Khudyakov, V. A. Kuzmin, and A. A. Ovchinnikov, Chem. Phys. Lett., 66, 565 (1979); K. Razi Naqvi, K. J. Mork, and S . Waldenstram, J . Phys. Chem., 84, 1315 (1980). (35) H. H. Schuh and H. Fischer, Helu. Chim. Acta, 61, 2130 (1978). (36) H. Hippler, K. Luther, A. V. Ravishankara, and J. Troe, Z . Phys. Chem. (Frankfurt am Main), 142, 1 (1984).
Elementary Reactions in Compressed Gases and Liquids
The Journal of Physical Chemistry, Vol. 90, No. 3, I986
361
the transition into the liquid phase, the wavelength dependence seems to diminish again as indicated in Figure 9 for iodine photolysis in liquid ethane. The experimental observations shown in Figures 4-7 and 9 suggest20the participation of two “mechanisms” restricting photolysis, a cage mechanism and a cluster mechanism. At first we consider only the cage mechanism. To a first approximation, one may try to treat the cage mechanism by a slowing-down stage of the separating fragments under the action of the surrounding medium and a subsequent diffusive separation in competition to in-cage energy relaxation. By solution of the corresponding diffusion equation one arrives at the
0
W
Figure 9. Quantum yields 6 for iodine photolysis in gaseous (open symbols) and liquid (filled symbols) ethane (photolysis wavelength X = 694 (squares) and 581 (circles) nm, measurements from ref 20; curve 1, eq 9; curve 2, eq 8; curves 3a and 3b, eq 14).
disappearance of methyl radicals is observed in the bath gas N2 at pressures at least one order of magnitude below the range where diffusion control sets in. Clearly, a new process here has to be taken into account. It was suggested36 that the intermediate formation of relatively long-lived N2CH3 radicals (with spectra similar to C H 3 which were monitored) slowed down the disappearance of (CH3 + N2CH3) radicals. A similar phenomenon was found3’ with the recombination of chlorine atoms in highpressure N 2 and C 0 2 . Again, the intermediate formation of relatively long-lived N2Cl and C02C1 radicals might be the explanation of the anomalously slow formation of C12. Such experiments indicate that chemical interactions between reactant radicals and “solventn molecules can become of importance in sufficiently dense environments. 2.4. Geminate Recombination and Photolysis Quantum Yields over Wide Density Ranges. Whereas photolysis from long-lived excited states is possible in low-pressure gases, it is generally quenched in dense solvents, see Figure 1. Photolysis from short-lived excited states may occur under the condition that cage effects do not induce a fast geminate recombination of the separating fragments. Time-resolved observations on a picosecond time scale of short-lived transients as well as measurements of photolysis quantum yields as a function of the solvent density reveal important aspects of the dynamics of the process. In the following, we only consider photolysis quantum yields since pressure dependences of the dynamics of the transient species involved have not been measured yet. The most complete set of observations available for the photolysis of diatomics is that of iodine20*29g38-41 and bromine.33 Quantum yields in these cases have been measured from low pressures up to 7 kbar in the gas phase (in some cases), they have been followed in liquid solvents up to 5.4 kbar. The effects of varying photolysis wavelength, temperature, and solvent nature have been studied systematically. Figures 4-7 and 9 include a series of experimental observations. Figures 4-6 for bromine photolysis in rare gases show that quantum yields $J are reduced to values below unity at pressures of the bath gases where diffusion control sets in. Obviously the onset of diffusion control and the formation of solvent cages around the photolyzing molecules are related phenomena. For more complex bath gases like N2 (Figure 7 ) and C 0 2 in bromine recombination, or ethane (Figure 9), propane, C 0 2 , and other bath gases in iodine recombination,20 one observes a change of this behavior. The quantum yields decline at pressures markedly below the turnover range of the recombination rate coefficients. The decay of $ occurs earlier, the longer the photolysis wavelength and the lower the t e m p e r a t ~ r e . At ~~ (37) H. Hippler and J. Troe, In?.J . Chem. Kine?., 8, 501 (1976). (38) K. Luther and J. Troe, Chem. Phys. Lett., 24, 85 (1974). (39) K. Luther, J. Schroeder, J. Troe, and U. Unterberg, J . Phys. Chem., 84, 3072 (1980). (40) M. Zellweger, and H. van den Bergh, J . Chem. Phys., 72, 5405 (1980). (41) C. Dupuy and H. van den Bergh, Chem. Phys. Lett., 57, 348 (1978).
with the same symbols as explained in eq 5-7, and ro denoting the average fragment distance where the initial excess energy has been lost. If ro < R , R / r o should be set equal to unity. Simple expressions for ro have been derived.*O They show that, for typical photolysis wavelengths, there is hardly any cage breakout (Le. ro < R ) for densities where eq 8 leads to #
