J. Phys. Chem. 1985,89, 5709-5713 N rotations in the course of a concerted action. The above picture implies that HT and HP structures be, in fact, identical. Since lattice energy calculations do not provide any hint on a second metastable molecular arrangement, this appears to be a plausible conclusion. In fact, all experimental evidence available so far is compatible with this interpretation except fluorescence spectra. Kalinowski et a1.28found that the fluorescence of the HP phase is red-shifted compared to that of the L T phase. However, for the following reason we disregard this observation as evidence against the identity of both phases: (i) TC fluorescence spectra show large inhomogeneous broadening, possibly due to defects. In addition, in the temperature range investigated the 0-0 emission band is subject to reabsorption effects.28 Therefore, the peak of the 0-0 band is not a good indicator for the energetic location of the emitting state. (ii) Pressure may well create additional structural traps for singlet excitons. Fluorescence spectra nevertheless indicate that there is a Stokes shift of order 200 cm-’ between absorption and emission. Occurrence of the asymmetric rotational displacement of molecule I1 provides a straightforward interpretation of this effect in terms of site relaxation of molecule 11. The driving force for the motion is the gradient of the intermolecular potential of an excited pair of molecules which favors a coplanar molecular arrangement. This is equivalent to self-trapping of a singlet exciton by coupling to a librational mode of molecule 11. Since the ground state of the trapping conformation is repulsive, the binding energy of the exciton is only a fraction of the energy loss between absorption and emission and the high mobility of singlet excitons (28) Kalinowski, J.; Jankowiak, R.; Bassler, H. J . Lumin. 1981, 22, 397.
5709
in the HT phasez9 is not in contradiction to the localized concept.
Concluding Remarks This work demonstrates that the analysis of fluorescence resonances in a magnetic field can profitably be used for structural analysis in cases where a full crystallographic study is inaccasible. Bearing in mind that delayed fluoresence resulting from the mutual annihilation of two triplet excitons can also be modulated by a magnetic field,’ it is by no means restricted to crystalline tetracene where one can monitor fission of the singlet exciton. A promising application of this method is the determination of the orientation of guest molecules doped into molecular crystals. Even if the dopant enters the host matrix substantially, it need not and, in fact, does not normally have the same orientation as the molecule it replaces. Examples are tetracene doped into a n t h r a ~ e n e ~ ~ , ~ ’ or pentacene in t e t r a ~ e n e . Analyzing ~~ the resonance pattern of heterofission of singlet excitons or of heterofission of triplet excitons appears to be a relatively simple tool for unravelling the packing characteristics of the guest molecule. The disadvantage of the method is that it is insensitive to translational molecular displacements.
Acknowledgment. Financial support by the Fonds der Chemischen Industrie is gratefully acknowledged. Registry No. TC, 92-24-0. (29) Campillo, A. J.; Hyer, R.; Shapiro, S . C.; Swenberg, C. E. Chem. Phys. Lett. 1917, 48, 495. (30) Ramdas, S . Chem. Phys. Lett. 1979, 60, 320. (31) Dorner, H.; Hundhausen, R.; Schmid, D. Chem. Phys. Lett. 1978,53, 101. (32) Jankowiak, R.; Kalinowski, J. Mol. Cryst. Liq. Cryst. 1978,48, 187.
Backward Electron Transfers wlthin a Geminate Pair Formed in the Quenching of the Phosphorescent States of Rhodium(I I I ) Compounds Takeshi Ohno Chemistry Department, College of General Education, Osaka University, Toyonaka, Osaka 560, Japan (Received: April 12, 1985)
Electron transfer quenching of the phosphorescent states of rhodium(II1) compounds was studied by nanosecond laser photolysis-kineticspectroscopy. Six aromatic amines and three methoxybenzenesin the mixed solvent of water and acetonitrile were oxidized by both the T-T* triplet excited states of rhodium(II1) tris(4,7-diphenyl-l,IO-phenanthroline) and rhodium(II1) tris( 1,lO-phenanthroline)and the ligand field triplet excited states of dichlororhodium(II1)bis(4,7-diphenyl- 1,IO-phenanthroline) and dichlororhodium(II1) bis( 1,lO-phenanthroline). The rates of spin-invertedbackward electron transfers within the geminate radical pair were derived from the efficiencies of the electron transfer product formation in the quenching. The backward electron transfer rates, which are not limited by the diffusion-controlled rate, gradually increased with the exothermicity but any “inverted behavior” did not appear in the highly exothermic region (-1.73 eV). Nonadiabaticity in the backward electron transfer is reflected in the weak dependence of the rate on the exothermicity.
Introduction Electron transfer is one of the most fundamental of chemical reactions. Reaction rates for bimolecular electron transfers in condensed media have been studied theoretically and experimentally. Adiabatic predicted that the reaction rates of electron transfer increase with exothermicity to a maximum and then decrease with increasing exothermicity in the highly exothermic region. A number of experimental studies of bimolecular electron transfer in solution3-’ have found that a collisional (1) Marcus, R. A. J. Chem. Phys. 1964, 43, 679-701. Annu. Rev. Phys. Chem. 1964, 15, 155-96. ( 2 ) Siders, P.; Marcus, R. A. J . Am. Chem. Sot. 1981, 103, 741-7. Marcus, R. A.; Siders, P. J . Phys. Chem. 1982, 86, 622-30. (3) Rehm, D.; Weller, A. Ber. Bunsenges. Phys. Chem. 1969, 73, 834-9.
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process limits the reaction rate before it reaches a proper maximum in the highly exothermic region. Fast electron transfer processes compared to the diffusion process in liquid solution have been measured in the backward reaction within a geminate radical pair which was generated in a bimolecular electron transfer of excited molecules.8-’z The rates of the fast backward electron transfer ~~~
(4) Kikuchi, K.; Tamura, S.; Iwanaga, C.; Kokubun, H.; Usui, Y. Z . Phys. Chem. (Frankfurt am Main) 1917, 106, 7-18. ( 5 ) Ballardini, R.; Varani, G.; Indelli, M. T.; Scandola, F.; Balzani, V. J . Am. Chem. SOC.1978, 100, 7219-23. (6) Nagle, J. K.; Dressick, W. J.; Meyer, T. J. J. Am. Chem. SOC.1979, 101, 3993-5. (7) Indelli, M. T.; Ballardini, R.; Scandola, F. J . Phys. Chem. 1984, 88, 2547-5 1 . ( 8 ) Masuhara, H.; Mataga, N. Acc. Chem. Rev. 1981, 14, 312-8.
0 1985 American Chemical Society
5710 The Journal of Physical Chemistry, Vol. 89, No. 26, 1985
Ohno
were derived from the formation efficiency of the free radical in TABLE I: Quenching Rate Constants (k,)and Efficiencies of the quenching of the excited state. We have studied backward Product Formation (F,)in AN-Water (1:l) electron transfers within geminate radical pairs which were k,/109 produced in the quenching of the phosphorescent ~ t a t e . ~ J ’ , ~ ~ reductants excited states M-’ s-’ F, -AGho/eV Reaction rates for spin-inverted electron transfers, which are DPA Rh(dp-phen)’’’ 9.7 0.38 1 73 heavily affected by electronic overlap and spin-orbit interaction,13 0.55 155 3,3’-DMB Rh(dp-~hen),~+ 8.4 have been found to be less dependent on the exothermi~ity.~J’J~ 1,2-PD Rh(dp-phen)? 8.1 0.64 1.44 In this paper, the efficiencies of electron-transfer product formation TMB Rh(dp-phen)’” 0.58 1.33 in the quenching of the phosphorescent states of rhodium(II1) 6.3 0.78 1.08 1,4-PD Rh(dp-pher&’+ compounds will be described. The effect of exothermicity on the TMPD Rh(dp-phen)’’+ 0.88 0.92 3,3’-DMB RhC12(dp-phen)2t 8.8 0.51 rates of backward electron transfer within the geminate pair in the highly exothermic region will be discussed in terms of an TMB+s,~O 32000 M-’ cm-’ at 460 nm for ~ , ~ - D M B +8860 V , ~ ’M-’ electronic interaction. Rhodium(II1) compounds of 1,locm-’ at 450 nm for 1,2,4-TMB+.,?29540 M-’ cm-I at 460 nm for phenanthroline (phen) and 4,7-diphenyl- 1,lO-phenanthroline ~ , ~ - D M B +and S ? 5710 ~ M-’ cm at 580 nm for 1,3,5-TMB+..22 (dp-phen) were studied by nanosecond laser photolysis specFormations of 1,2-PD+. and DPA’. were determined after hole troscopy. A couple of studies on the photochemical redox reactions transfer from them to 1,4-PD or 3,3’-DMB which has a lower of rhodium(II1) compounds have been r e p ~ r t e d . ~ . ’ ~ redox potential than 1,2-PD and DPA. The redox potential of Experimental Section R h ( d ~ - p h e n ) ~was ~ + obtained as a cathodic half-peak potential Apparatus. An N E C SLG-2018 Q-switched ruby laser capable (Ep,2)of the fresh acetonitrile solution containing 0.1 M tetraof providing up to 0.3 J per flash at 347 nm was used. The details butylammonium tetrafluoroborate, though the triangular wave of the laser apparatus and the monitoring device have been devoltammogram was irreversible. scribed elsewhere.12J5 Transient changes in the absorption were fed into a Iwatsu 8123 Storagescope, followed by computer Results analysis using a NEC PC8001 MKII microcomputer. A xenon Electron Transfer Quenching of the Triplet Excited State of flash photolysis apparatus was used to excite R h ( d ~ - p h e n ) ~in~ + Rh(dp-phen)33+by Aromatic Amines. Laser excitation of Rhthe presence of T M B and TMPD which is photoionized by the ( d ~ - p h e n ) ~in~ AN-water + (1:l) gave rise to production of a 347-nm light of the ruby laser. The details of the flash photolysis transient absorption with a maximum at 550 nm, which is in have been described e1~ewhere.l~~ Electrochemical measurements agreement with the absorption spectrum of the phosphorescent were made with the help of Dr. Morishima of Osaka University. state.I2 The phosphorescent state localized in the ligand of dpphen The details of his apparatus are described elsewhere.I6 with a lifetime of 100 ps was quenched by the following aromatic Materials. Preparations and purifications of [Rh(phen),]compound with redox potentialz3given in parentheses: DPA (0.83 C13.3H20, [Rh(dp-phen)3](C104)34H20, [RhC12(phen)2]C1. V), 3,3’-DMB (0.65 V), 1,2-PD (0.54 V12), TMB (0.43 V), 1,4-PD 3 H 2 0 , and [RhC12(dp-phen)2]C1+3Hz0 were previously de(0.18 V), and TMPD (0.02 V). Production of an intense abscribed.” Aromatic amines, N,N,N’,N’-tetramethyl-1,4- sorption by the oxidized quencher was spectroscopically observed, phenylenediamine (TMPD), 1,4-~henylenediamine(1,4-PD), while the reduced Rh(II1) species was not measured because -f N,N,N’,N’-tetramethylbenzidine (TMB), and 1,2-phenylenediits weak a b s o r p t i ~ n . Nearly ~~ diffusion-controlled rate constant amine (1,2-PD) were purified by recrystallization and vacuum for electron transfer quenching were obtained by plotting the dec,l> sublimation. Diphenylamine (DPA), 3,3’-dimet h ylbenzidine rate constants for the excited-state absorption against the con(3,3’-DMB), 1,2,4-trimethoxybenzene (1,2,4-TMB), 1,4-dicentration of quencher, the values of which are given in Table methoxybenzene (1 ,4-DMB), and 1,3,5-trirnethoxybenzene I. The excited-state absorption of RhC12(dp-phen)2+ with a (1,3,5-TMB) were used as supplied by Tokyo Kasei Co. Acelifetime of 200 ns in AN-water ( l : l ) , which has been identified tonitrile (AN) was also used as supplied by Wako Pure Chemical as a triplet excited-state localized in the metal center,I7 was Co. Water was purified by passing it through a Millipore deionizer quenched by 3,3’-DMB with 8.8 X lo9 M-I s-l. Formation of the and filter. A mixed solvent of A N and water (1:l) with no cation radical of the quencher was spectroscopically observed in electrolyte was used unless otherwise noted. The test solutions the quenching of the excited states of R h ( d ~ - p h e n ) , ~and + were deaerated by purging for 15-20 min with nitrogen. RhC12(dp-phen)2+ as in the quenching of the excited state of Measurements. All the measurements were done at 15 f 2 Rh(phe~~)~~+ Efficiencies .’J~ for electron-transfer product forOC. Production of excited rhodium(II1) compounds was specmation in the quenchings of 3Rh(dp-phen)33+and 3RhC12(dptroscopically measured by using the molar absorption coefficients phen),’ were obtained as a ratio of radical formation to triplet of the triplet excited state^,'^^'^ 9800 M-’ cm-’ at 550 nm for formation in the complete quenching of the triplet excited state, R h ( d p ~ h e n ) ~2300 ~ + , M-l cm-’ at 510 nm for R h ( ~ h e n ) ~2750 ~+, as Table I shows. The triplet excited state and the radical proM-’ cm-’ at 560 nm for RhC12(dp-phen)2+,1600 M-’ cm-l at 550 duced were determined from their molar absorption coefficients. nm for RhClz(phen)z+. The formation of the cation radicals of Since TMB and TMPD undergo photoionization on 347-nm exthe organic quenchers was also determined by transient speccitation, the efficiencies were determined by selective excitation troscopy. The molar absorption coefficients of the cation radicals of the rhodium(II1) compound by using a xenon flash apparatus utilized are 12000 M-’ cm-l at 610 nm for TMPD+.,19 7000 M-’ with 370-nm cutoff filter. 3Rh(dp-phen)33+in the presence of cm-I at 460 nm for 1,4-PD+.,19 41 000 M-’ cm-’ at 474 nm for TMB or TMPD (20 pM) was not observed because of the longer duration of the xenon flash. (9) Ohno, T.; Lichtin, N. N. J . Phys. Chem. 1982,86, 354-60. Electron Transfer Quenchings of the Triplet Excited States (10) Iwa, P.; Steiner, U. E.; Vogelmann, E.; Kramer, H. E. A. J . Phys. of the Rhodium(I1I) Compounds by Methoxybenzenes. ExcitChem. 1982,86, 1277-85. (1 1) Ohno, T.; Kato, S.; Yamada, A.; Tanno, T. J . Phys. Chem. 1983,87, ed-state absorptions of RhC12(dpphen)2+in AN-water (1:l) and 775-8 1. RhC1z(phen)2+ in AN-water (4:1), which have been assigned as (12) Ohno, T.; Kato, S. Bull. Chem. SOC.Jpn. 1984, 58, 1528-33. a ligand-field excited state localized on the rhodium(II1) (3LF),17 (13) (a) Winter, G.; Steiner, U. Ber. Bunsenges. Phys. Chem. 1980, 84, 1203-14. (b) Ohno, T.; Kato, S. J . Phys. Chem. 1984,88, 1670-4. (14) (a) Kirch, M.; Lehn, J.-P.Helu. Chim. Acta 1979,62, 1345-84. (b) Ballardini, R.; Varani, G.;Balzani, V. J. Am. Chem. Soc. 1980,102, 1719-20. (15) Ohno, T.; Kato, S.; Lichtin, N. N. Bull. Chem. SOC.Jpn. 1982, 55, 2753-9. (16) Morishima, Y.; Itoh, Y.; Koyagi, A. J. Polym. Sci. Polym. Chem. Ed. 1983, 21, 953-60. (17) Ohno, T.; Kato, S. Bull. Chem. SOC.Jpn. 1984, 57, 3391-4. (18) Ohno, T. Coord. Chem. Reu. 1985, 64, 311-20. (19) Rao, P. S.;Hayon, E. J. Phys. Chem. 1975, 79, 1063-6.
(20) Akaitis, S . A.; Gratzel, M. J. Am. Chem. SOC.1976, 98, 3549-54. (21) Ohno, T., unpublished work. (22) O’Neile, P.; Steenken, S.; Schulte-Frohlinde, G. J. Phys. Chem. 1975, 70 . , 2777-9 - - . (23) Mann, C. K.; Barnes, K. K. “Electrochemical Reactions in Nonaqueous Systems”; Marcel Dekker: New York, 1970. (24) Mulazzani, Q.G.; Emmi, S.; Hoffman, M. Z.; Ventuni, M. J . Am. Chem. SOC.1981, 103, 3362-70.
The Journal of Physical Chemistry, Vol. 89, No. 26, 1985 5711
Back Electron Transfer within a Geminate Pair
TABLE II: Quenching Rate Constants (k4/109 M-' s-') and Efficiencies of Product Formation (F,)in AN-Water (1:l)
1,3,5-TMB (1.49 V y 1,4-DMB (1.34 V y 1,2,4-TMB (1.12 V)
Rh(dp-~hen)~'+, E. = 2.59 e W b
RhCI2(dp-phen),+, E. = 2.1 e W C
4.0 4.0
~the~gradual ~ * ~ increase ~ , ~ ~in the bimolecular electron transfer rates with exothermicity is rather consistent with the decrease in F , observed in this work. As Figure 1 shows, the rate of backward electron transfer continues to increase with the exothermicity and the dependence of log kb on AGbo is weakly negative (-1.2 v-’). The weak dependence of log kb upon AGbo, which was obtained for electron transfer between the Rh(I1) compound and the amine cation radicals over a wider range of exothermicity than so far studied, indicates nonadiabaticity in the backward electron transfer within the geminate pair. Unless (33) Creutz, C.; Keller, A. D.; Sutin, N . ; Zipp, A . P. J . Am. Chem. SOC. 1982, 104, 3618-27. Creutz, C.; Keller, A. D.; Schwartz, H. A.; Sutin, N . ; Zipp, A. P. ACS Symp. Ser. 1982, No. 198, 385-96. (34) Kew. G.; DeArmond, K.; Hanck, K. J . Phys. Chem. 1974, 78,727-34.
Ohno unpaired electrons within the geminate radical pair suffer a spin-orbit interaction through electronic overlap, electron transfer is forbidden by the spin-conservation rule. Consequently, one may explain the weak dependence of kb on AGbo (d log kb/dAGbo = -1.2 V-’) for backward electron transfer between Rh(d~-phen),~+ and the cation radical of the aromatic amine, since the odd electron in the do* of R h ( d p - ~ h e n ) ~is~shielded + by the bulky ligands from interacting with the n orbital of the cation radical. If not Rh(II1) but a ligand of dp-phen is reduced in the excited-state reaction,,, the weak dependence of k b on AGbo can be explained in terms of the weak spin-orbit interaction in the radical pair. Lack of spin-orbit interaction brings agout a weak dependence (d log kb/dAGbo = -1.0 v-’) in the electron transfer within the geminate radical pair formed in the quenching of the triplet excited organic compound by aromatic amine^.^^^' This situation is consistent with the fact that the rates of spin-inverted electron transfer are more dependent on AGbo (d log kb/dAGbo = -2.4 V-I) for electron transfer which occurs (1) from the d a orbital of chromium(I1) to the n orbital of the aromatic amine radical,12 and (2) from the n orbital of the aromatic amine radical to the d a orbital of iron (III).~ Another explanation for the gradual increase of kb with exothermicity is that the reaction mechanism changes with exothermicity from an outer-sphere electron transfer within the geminate radical pair to an intramolecular radiationless transition in the charge-transfer complex formed between radicals. The Franck-Condon factor for the radiationless transition is expected to follow the “energy gap law”35when intermolecular electron transfer occurs as a radiationless transition between weakly interacting states having no potential crossing.8 Backward Electron Transfer within the Geminate Pair Formed in the Quenching of Triplet Excited States by Methoxybenzenes. The phosphorescent states of the rhodium(II1) complexes were quenched by the methoxybenzenes with a rate faster than lo8 M-’ s-’. The Gibbs enthalpy change (AG,) for the quenching of the excited rhodium(II1) complexes are calculated from the redox potentials of the rhodium(II1) complex and the methoxybenzene, the excitation energy of the complex, and the electrostatic energy of the ionic product. A AG, more negative than -0.19 eV for the quenching of ,Rh(d~-phen)~,+ and 3Rh(phen)33+by the methoxybenzenes is consistent with the nearly diffusion-controlled rate. Though the redox potentials of [RhC12(dp-phen)2]+ and [RhC12(phen)2]+are unknown, the energies of the lowest L F excited states of the dichloro c ~ m p l e x e s ,which ~ ~ , ~are ~ estimated to be lower than 2.2 eV from the onset of the broad L F phosphorescence as shown in Table 11, are responsible for the quenching rate constants smaller than the diffusion-controlled values. The radical formation efficiencies af the quenching events by the methoxybenzenes are in the order of R h ( d ~ - p h e n ) , ~> + [RhClz(dpphen)2]+2 Rh(phen),,+ > [RhCl*(~hen)~]+, while the data are not sufficient for comparison. The rates of two processes, the backward electron transfer and the dissociation of the geminate pair, are variants in this case. The backward electron transfer rate within the geminate pair is dependent of the AGb (the redox potentials of the reactants) and the electronic interaction. The ligands, dp-phen, phen, and C1-, may affect the electronic overlap between the do* electron of the rhodium(I1) and the nonbonding electron of the methoxybenzene radical, which is indispensable for both electron transfer and spin inversion. It is probable that the largest ligand, dp-phen, may retard the spin inversion from occurring through electronic overlap between the do* and the nonbonding electrons as the dp-phen of Cr(d~-phen),~+ reduced the kb more than the phen of C r ( ~ h e n ) , ~did.I2 + The efficiencies of 1,4-DBM+. and 1,3,5-TMB+. formation are larger than that of DPA+. formation (0.38). In other words, the rate of backward electron transfer between R h ( d ~ - p h e n ) ~and ~+ 1,4-DMB+. or 1,3,5-TMBC-with AGb more negative than -2.2 (35) Henry, B. R.; Siebrand, W. In “Organic Molecular Photophysics”, Vol. 1 , Birks, J. B., Ed.; Wiley: New York, 1973. (36) Carstens, D. H. W.; Crosby, G. A. J . Mol. Specfrosc. 1970, 34, 113-35.
J. Phys. Chem. 1985, 89, 5713-5719 eV is slower than that between R h ( d p - ~ h e n ) ~and ~ +DPA’. with a AGb of -1.7 eV. The slower rates may not be ascribed to a smaller isoenergetic vibrational overlap but to a smaller electronic interaction between Rh(dp-phen)32+ and the methoxybenzene radical in the backward electron transfer.
Acknowledgment. The author thanks Dr. Y.Morishima for his help with the measurement of the redox potential and Dr. A.
5713
Yoshimura for his help with the computer programing. This research was supported in part by the Japanese Ministry of Education, Science and Cultures. Registry No. TMPD, 100-22-1; 1,4-PD, 106-50-3; TMB, 20627-78-5; 1,2-PD, 95-54-5; DPA, 122-39-4; 3,3’-DMB, 119-93-7; 1,2,4-TMB, 9563-6; 1,4-DMB, 150-78-7; 1,3,5-TMB, 621-23-8; [Rh(dp-phen)Jj+, 94552-8 1-5; [Rh(phen),13+, 47837-61-6; [RhCl,(dp-phen),]*, 9578743-2; [RhCl,(phen),]+, 52950-59-1.
Sequential Impulse Model for Chemical Reactions: A Comparison of the Reactive Scattering Contour Diagrams for Limiting Cases S. A. Safron Chemistry Department, Florida State University, Tallahassee, Florida 32306-3006 (Received: June 5. 1985)
The sequential impulse model for chemical reactions is developed in a somewhat more general and “realistic” form than previously presented. The differential cross sections for the scattered products are given in terms of elliptical-type integrals. Four different cases are considered for the same sequence of collisions, A strikes B and then B strikes C: for the reaction A + BC AB + C, energy may be released/absorbed in either the first or second collision; for the reaction A + BC AC + B (knockout reaction), energy may be released/absorbed in the first or second collision. Each case is shown to lead to contour diagrams for the scattering, which have characteristic shapes and velocity scaling.
-
-
Introduction Mahan, Ruska, and Winn‘ have shown that the sequential impulse model (SIM) is a useful description of the dynamics of the chemical reaction A
+ BC -,AB + C
(1)
when the mechanism can be approximated as a hard-sphere impulsive collision between A and B followed by a similar impulsive collision between B and C. These authors examined the velocity space geometry in the center-of-mass (c.m.) frame when both of the collisions are elastic and obtained the SIM prediction for the differential cross section of the scattered AB product in the form of a relatively straightforward integral expression. Combining the range of A-B impact parameters with the possible BC orientations that lead to scattering at center-of-mass velocity uAB within volume element d3uABdefines a particular locus of points in velocity space; the differential cross section is proportional to the integral over these points. (The absolute differential cross section depends on the A-B collision diameter or cross section.) For a given reaction system, the locus depends on only one free parameter, the ratio of the equilibrium separation between B and C in the BC molecule to the “hard-core” collision diameter between B and C. In general terms, with the elastic collision assumption, the model’ predicts that the reactively scattered products should be confined to within a cardioid-shaped boundary in velocity space and that the region of greatest product flux is in the immediate neighborhood of the “spectator stripping” velocity.* Ignoring the aphorism that “Experimenters operate in the laboratory frame, but physics operates in the center-of-mass frame”, Safron, Coppenger, and Smith3 decided to reconsider this model from the point of view of sequential collisions in the laboratory (LAB) frame with BC initially stationary. By requiring the ratio parameter to be unity, they were able to obtain simple (1) B. H. Mahan, W. E. W. Ruska, and J. S. Winn, J. Chem. Phys., 65, 3888 (1976). (2) For example, see R. D. Levine and R. B. Bernstein, “Molecular Reaction Dynamics”, Oxford University Press, New York, 1974. (3) S.A. Safron, G. W. Coppenger, and V. F.Smith, J . Chem. Phys., 80, 1929 (1984).
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analytical expressions for the differential cross section even when the collisions are not elastic. In particular, for endoergic reactions they found that the boundaries for the scattering in velocity space and the general appearance of the scattering flux contour diagrams ought to be very different depending on whether the energy of reaction is absorbed in the A-B collision or in the B-C collision. They further found that they could use the S I M to obtain differential cross sections for the “knockout” reaction4 A
+ BC -,AC + B
(2)
which takes place following the same sequence of collisions; that is, B is knocked out by A and glances off C as it depart^.^ This treatment of the S I M is extended here to allow for ratios of BC equilibrium separation to hard-core diameter which are not unity. In addition, examples of the predicted reative scattering contour diagrams for exo- and endoergic reactions are presented and their features compared and discussed. We find for reaction 1, as Mahan and co-workers did, that the general shape of the contour diagram for the scattering is not greatly affected by varying the ratio parameter. However, more interestingly, for reaction 2 a ratio differing just slightly from unity greatly diminishes the very steep sideways “ridge” in the contour diagram reported previ~usly,~ which appears to arise from a kind of “edge effect” due to the unrealistically sharp cutoffs inherent in hardsphere scattering. This result should moderate with caution the import of another aphorism that “The behavior of hard spheres reveals the essence without any of the mess”.
Hard-Sphere Sequential Impulse Model The treatment by Safron et ale3s4of the S I M is based on the set of assumptions: (i) the reaction takes place as the result of the sequence of hard-sphere collisions, first A strikes B and then B strikes C; (ii) the scattering is due to an impulsive force directed (4) S. A. Safron and G. W. Coppenger, J . Chem. Phys., 80,4907 (1984). ( 5 ) The experiments reported by S. A. Safron, G. A. King, and R. C. Horvat, J . Am. Chem. SOC.,103, 6333 (1981), suggest that the reactions are of the knockout type. See S. A. Safron, J. Phys. Chem., following article in this issue, where the model presented here is shown to give a reasonable fit to these reported data.
0 1985 American Chemical Society
