15162
J. Phys. Chem. 1996, 100, 15162-15164
Picosecond Kinetic Study of the Photoinduced Homolysis and Heterolysis of Diphenylmethyl Bromide. 2. Role of Polarization Caging in Contact Ion Pair Recombination Jens Dreyer, Matthew Lipson, and Kevin S. Peters* Department of Chemistry and Biochemistry, UniVersity of Colorado, Boulder, Colorado 80309-0215 ReceiVed: April 25, 1996; In Final Form: July 8, 1996X
Picosecond absorption spectroscopy is used to study the dynamics of the collapse of the diphenylmethyl bromide contact ion pair to form the carbon-bromine bond. The dynamics of the diphenylmethyl cationbromide anion are probed as a function of temperature. The Arrhenius parameters for the collapse of the contact ion pair to form the carbon-bromine bond are A ) 4.9 × 1011 s-1 and Ea ) 3.0 kcal/mol. Deviations from equilibrium solvation transition state theory are found, and it is concluded that the collapse of the ion pair occurs in the polarization caging regime.
Introduction The second part in this two-paper series examines the dynamics of the carbon-bromine bond formation resulting from the collapse of the diphenylmethyl bromide contact ion pair. An analogous study of the dynamics of carbon-chlorine bond formation has been presented.1 The kinetics of the collapse of the diphenylmethyl chloride contact ion pair to form the C-Cl bond in acetonitrile and propionitrile has been examined as a function of temperature, from which the Arrhenius parameters were derived. Analysis of the activation parameters reveals that the kinetics for covalent bond formation deviates from the predictions made by classical transition state theory and that the collapse of the contact ion pair occurs within the polarization caging regime.2 In this regime, as the charged system attempts to move off the transition state, it finds itself trapped in a solvent well or a “polarization cage”, and it is only when the solvent cage relaxes that the system may evolve into product. Thus the evolution into product is controlled by solvent dynamics, as evidenced by the A factor solvent dependence,1 and the rate of the reaction is less than that predicted by transition state theory, which assumes that the solvent instantaneously responds to the motion of charge in the transition state. The main focus of the present study will be to determine the activation parameters for the collapse of diphenylmethyl bromide contact ion pair forming the carbon-bromine (C-Br) covalent bond and to ascertain if the activation parameters for this process deviate from those predicted by transition state theory. These findings will be compared with those obtained in our prior study of C-Cl covalent bond formation. We find that although the A factors for the two systems differ by a factor of 2, the magnitude of the deviation from the prediction made by transition state theory is similar for the two systems, suggesting that the collapse of the diphenylmethyl bromide contact ion pair also occurs in the polarization caging regime. Furthermore, the difference in the equilibrium potential of mean force between the contact ion pair and the transition state is the same within the error of the experiment for the two systems. Experimental Section Diphenylmethyl bromide was obtained from Aldrich and purified by sublimation. The purity, >95%, was measured by gas chromatography (HP5890,FID,DB17 column) and by NMR. X
Abstract published in AdVance ACS Abstracts, September 1, 1996.
S0022-3654(96)01200-2 CCC: $12.00
Figure 1. Arrhenius analysis of the temperature dependence of k1 and k2 for the decay of the diphenylmethyl bromide CIP in acetonitrile.
TABLE 1: Arrhenius Parameters for Decay of the DPMB CIP and DPMC CIP in Acetonitrile ln A (s-1)
A (s-1)
k1 k2
26.92 ((0.44)a 25.97 ((0.49)a
DPMB 4.9 × 1011 1.9 × 1011
k1 k2
27.55 ((0.53)a 27.20 ((0.94)a
DPMCb 9.2 × 1011 6.5 × 1011
a
Ea (kcal/mol) 3.0 ((0.26)a 2.1 ((0.29)a 3.2 ((0.32)a 3.1 ((0.56)a
Error represents 1σ. b From ref 1.
Acetonitrile, Burdick and Jackson-UV-grade, was distilled from calcium hydride. A description of the picosecond absorption spectrometer and the method for deconvolution of the kinetic data has been presented in the preceding paper.3 Results The temperature dependence for the decay of the diphenylmethyl bromide contact ion pair (DPMB CIP) was examined over the temperature range 8.8-52.4 °C. An Arrhenius analysis of the kinetic data for k1 and k2, defined in Scheme 1 in the preceding paper,3 is shown in Figure 1. The activation parameters for the decay of the DPMB CIP are given in Table 1 as well as the activation parameters for the decay of the diphenylmethyl chloride contact ion pair (DPMC CIP) obtained previously.1 © 1996 American Chemical Society
Photoinduced Homolysis and Heterolysis of DPMB. 2 Discussion The Theory of Polarization Caging. For reactions that involve the displacement of charge, the solvent is fundamental in controlling the reaction dynamics. The solvent effect is manifest in both a static and a dynamic manner. The static component is integral in effecting the potential of mean force for the reaction through the solvation of the reactants, transition state, and products. The dynamic component is revealed through the coupling of the motion of charge through the transition state with the solvent motion. For the dynamic component two limiting situations are envisioned. In the first limit, that of frozen solvation, the charge system passes through the transition state with no reorganization of the surrounding medium. In the second limit, that of equilibrium solvation, the solvent fully equilibrates with the motion of charge through the transition state. Reactions intermediate between the two extremes occur under conditions of nonequilibrium solvation. The question that arises is which parameters of the reacting system determine whether the reaction occurs under conditions of full solvation or nonequilibrium solvation. Hynes and co-workers, employing the generalized Langevin equation, have developed a theoretical formulation that explicitly addresses the nature of the coupling between reactant and solvent.2,4 The critical parameters contained in the theory are the parabolic reaction barrier frequency at the transition state, ωb, and a time-dependent friction coefficient, ζ(t), which describes the time-dependent interaction of the solvent with the displacement of charge in the transition state.5 From ζ(t) an electrostatic solvent frequency ωs is defined, which is a measure of the nondissipative restoring force that the charge system experiences as it attempts to move off the transition state. Finally the time scale for the solvent relaxation about the charge system τl, the longitudinal relaxation time, is critical in determining the reaction dynamics. Whether the reaction occurs under the conditions of full solvation or nonequilibrium solvation is determined by the relationship between the transition state reaction barrier frequency, ωb, and the longitudinal relaxation time of the solvent, τl. If ωb is sufficiently small so that condition ωbτl , 1 exists, then as the reacting system passes through the transition state, the solvent fully equilibrates to the motion of charge and the passage of the charge species through the transition state is unimpeded by the motion of the solvent. This is a fundamental assumption of transition state theory, and the rate constant for this limit is kTST. This assumption breaks down, however, when ωb increases relative to τl. When the polarization relaxation time of the solvent τl begins to lag the motion of charge through the transition state, a friction develops which serves to impede the motion of charge, thus reducing the rate of the reaction, k, below the value of kTST, so that the ratio of the two rate constants, κ ) k/kTST, is less than 1.0. The magnitude for the deviation of the reaction rate from that predicted by transition state theory, κ, depends critically upon the electrostatic solvent frequency, ωs. When the absolute magnitude of ωs is greater than the reaction barrier frequency ωb, |ωs| > |ωb|,as the charged species moves off the transition state toward product formation, it finds itself trapped in a solvent “polarization cage”, which prevents the continued motion toward product. It is only with the relaxation of the solvent cage, on a time scale τl, that the system can evolve toward product; while trapped in the polarization cage, the reactants undergo an oscillatory motion in the transition state, recrossing the barrier top. Thus the reaction kinetics is controlled by the dynamics of the relaxation of the solvent. Since transition state theory assumes that each crossing of the transition state leads to product formation, the effect of the polarization cage is to reduce the
J. Phys. Chem., Vol. 100, No. 37, 1996 15163 rate reaction constant, k, below that predicted by transition state theory, kTST. In the model study of van der Zwan and Hynes,2 when ωs is a factor of 4 larger than ωb, the limiting value of κ approaches 0.1, so that the rate of the reaction is an order of magnitude less than that predicted by transition state theory. When the absolute magnitude of the electrostatic solvent frequency ωs is less than the reaction barrier frequency ωb, |ωs| < |ωb|, as the charge species move off the transition state toward product, it does not experience a polarization cage preventing its evolution toward product, but instead the system proceeds toward product without the solvent having to rearrange. However the reacting species does feel a retarding force or a drag as a result of the electrostatic interaction with the solvent, so that the rate of the reaction is again reduced relative to that predicted by transition state theory. In this regime, where there is nonequilibrium solvation of the reacting species in the transition state, the nonadiabatic solvation limit, the limiting value of κ is on the order of 0.8, so that the deviation from the prediction of transition state theory is not large.2 Whether a reaction occurs in the limit of polarization caging or nonadiabatic solvation depends, then, upon ωb, ωs and τl. Although τl can be approximated by experiment, presently methods do not exist for the determination of ωb and ωs, and therefore it is not possible to predict, for a given reaction, the effect that the solvent will have upon reducing the rate of the reaction below that predicted by transition state theory. The Determination of K for CIP Recombination. The assessment of the magnitude of the deviation from the predictions of transition state theory for the rate constant of the collapse of the diphenylmethyl bromide CIP requires the determination of κ, defined as κ ) k/kTST, where k is the experimentally obtained rate constant and kTST is the rate constant predicted by transition state theory, a quantity that can only be estimated. The value of kTST is obtained through the expression6
kTST ) (kBT/h) (Qs(r†) Qrot(r†)/Qvib(R) Qs(R) Qrot(R)) × exp(-∆G†/kBT) (1) where kB is the Boltzmann constant and T is the temperature. The term ∆G† represents the difference in the equilibrium potential of mean force for the CIP and the transition state (TS) for covalent bond formation. The quantity Qvib(R) is the vibrational partition function for the CIP associated with the reactive mode within the CIP. Its value is approximated by6
Qvib(R) ≈ kBT/pωR
(2)
where ωR is the vibrational frequency that leads to passage through the transition state. It is assumed that this reaction frequency is that associated with the ion pair stretching frequency. Unfortunately, there has been no experimental determination of the contact ion pair stretching frequency for the diphenylmethyl bromide CIP, and thus its value can only be estimated. In our previous determination of the value for κ for the collapse of the diphenylmethyl chloride CIP, the reaction frequency ωR was estimated to be 150 cm-1; this estimate was based upon the vibrational frequencies of the alkali ion pair vibrations.1 An estimate of the diphenylmethyl bromide CIP stretching frequency can then be obtained from the diphenylmethyl chloride stretching frequency if it is assumed that the frequencies scale by the square root of the ratio of the reduced masses for the two ion pairs, leading to a value of 110 cm-1 for the diphenylmethyl bromide CIP stretching frequency.
15164 J. Phys. Chem., Vol. 100, No. 37, 1996
Dreyer et al.
The ratio of the terms Qrot(r†)/Qrot(R) represents the ratio of the rotational partition functions for the TS and the CIP and is given by6
Qrot(r†)/Qrot(R) ) (IaIbIc)TS1/2/(IaIbIc)CIP1/2
(3)
where Ia, Ib, and Ic are the moments of inertia about the principal axes of the TS and the CIP. These rotational partition functions, as defined, are for gas phase entities and do not account for interaction with the solvent. Since it is not possible to experimentally determine the magnitude of the effect that the solvent has upon the rotational partition function for the transition state, it will be assumed that the solvent effects upon the rotational partition function of the TS and the CIP will cancel. The ratio of the products of moments of inertia, eq 3, can be obtained if the geometries of the transition state and CIP were known. From Hynes’ theoretical studies of the SN1 dissociation of tert-butyl bromide in acetonitrile, the position of the transition state occurs at a carbon-chlorine separation of 2.7 Å, which increases to 3.5 Å in the CIP.7 Employing these values and assuming the diphenylmethyl cation remains planar as the CIP collapses to form the transition state, the value of Qrot(r†)/Qrot(R) is 0.73. If sp2 f sp3 rehybridization of the diphenylmethyl cation occurs upon the CIP collapse, the ratio increases to 0.87. The final terms in eq 1, Qs(r†) and Qs(R), are the partition functions associated with solvation of the TS and the CIP, whose values should differ given the difference in the electronic structure of the TS and the CIP.8 As shown by Hynes,6 the ratio between the two solvation partition functions is obtained through the relation
Qs(r†)/Qs(R) ) ωs(R)/ωs(r†)
(4)
where ωs(R) is the solvent frequency associated with the CIP and ωs(r†) is the solvent frequency associated the TS. Again there are no experimental values for these quantities, and thus it is necessary to resort to the Hynes theoretical study of tertbutyl chloride, where they find that ωs(R)/ωs(r†) ) 1.5.8 The decrease in the solvent vibrational frequency on going from the CIP to the TS is due to the greater polarizability of the transition state relative to the CIP. This is attributed to the stronger electronic coupling between the two diabatic states at the position of the transition state, leading to a greater polarizability of the transition state relative to the CIP. Based on the above estimates, the rate expression, eq 1, for the collapse of the diphenylmethyl bromide CIP is
kTST ) 3.3 × 1012 (s-1) exp(-∆G†/kbT)
(5)
where the prefactor 3.3 × 1012 (s-1) is assumed to be temperature independent. Since the temperature dependence is manifested only in the exponential term, then the difference in the potential of mean force for the TS and CIP, ∆G†, is equated with the Ea obtained from experiment, so that ∆G†) 3.0 kcal/mol in acetonitrile. The transmission coefficient κ for the collapse of the CIP is then just the ratio of the prefactors from the Arrhenius equation and from the transition state theory rate expression,
κ ) (4.9 × 1011 s-1)/(3.3 × 1012 s-1) ) 0.15
(6)
In deriving the rate expression given by eq 5, several assumptions were made in evaluating the partition functions given in eq 1, and of particular concern is the estimate of the
ratio ωs(R)/ωs(r†). Thus it is informative to examine the effect the estimate for ωs(R)/ωs(r†) has upon the determination of κ. If this ratio is greater than 1.5, then the value κ is further reduced, leading to an even greater deviation from the prediction of transition state theory. If, however, the ratio is approximately 1.0, then the value of κ increases to 0.23, which still displays a significant deviation from the prediction of transition state theory. Within the context of Hynes’ theory that addresses the nature of the coupling between reactant and solvent, the collapse of the diphenylmethyl bromide CIP to form the C-Br bond falls within the polarization caging regime.2 In this regime, it is the dynamics of the solvent, with a relaxation time scale of τl, that controls the motion of the charged species through the transition state so that the covalent bond cannot be formed until there is a rearrangement in the solvent structure surrounding the ion pair. While in the region of the transition state, the system undergoes approximately six oscillations (1/κ) prior to the successful formation of the C-Br bond. In our previous study of the kinetics for the collapse of the diphenylmethyl chloride CIP in acetonitrile,1 the difference in the potential of mean force for the process was found to be ∆G† ) 3.2 kcal/mol, which is virtually identical to that found for diphenylmethyl bromide, ∆G† ) 3.0 kcal/mol. The difference in the potential of mean force for the collapse of the CIP contains contributions from both the electronic barrier and solvent reorganization barrier. Presumably the energy for reorganization of the solvent about the ion pair should be the same for both the chloride and bromide ion pairs. Since the energy of activation association with the τl of acetonitrile is on the order of 0.8 kcal/mol,1 the electronic barrier for the collapse of the CIP is then on the order of 2.2 kcal/mol. In the Hynes theory, the parameters for the system that control the dynamics of the collapse of the CIP in the transition state are the reaction barrier frequency at the transition state, ωb, the electrostatic solvent frequency, ωs, and the characteristic response time of the solvent to the charge, τl.2 Since the electronic barriers for collapse of the chloride and bromide CIP are virtually identical, then ωb should be similar for the two species. Thus, with all three parameters being identical for the two CIPs, the value κ should be identical for the two systems. Indeed, although the experimental A factors for the collapse of the CIP are 9.2 x 1011 s-1 for the diphenylmethyl chloride CIP and 4.9 × 1011 s-1 for the diphenylmethyl bromide CIP, the derived κ values for the chloride ion pair and the bromide ion pair are 0.16 and 0.15, respectively. Thus it appears that the ωb, ωs, and τl values for the collapse of the diphenylmethyl chloride CIP are virtually identical to those for the collapse of the diphenylmethyl bromide CIP. Acknowledgment. This work is supported by a grant from the National Science Foundation, CHE 9408354. J.D. is grateful to the Deutsche Forschungsgemeinschaft for a postdoctoral grant. References and Notes (1) (1) Deniz, A. A.; Li, B.; Peters, K. S. J. Phys. Chem. 1995, 99, 12209. (2) Zwan, G. v. d.; Hynes, J. T. J. Chem. Phys. 1982, 76, 2993. (3) Dreyer, J.; Peters, K. S. J. Phys. Chem. 1996, 100, 15156. (4) Zichi, D. A.; Hynes, J. T. J. Phys. Chem. 1988, 88, 2513. (5) Grote, R. F.; Hynes, J. T. J. Chem. Phys. 1980, 73, 2715. (6) Kim, H. J.; Hynes, J. T. J. Am. Chem. Soc. 1992, 114, 10528. (7) Mathis, J. R.; Kim, H. J.; Hynes, J. T. J. Am. Chem. Soc. 1993, 115, 8248. (8) Kim, H. J.; Hynes, J. T. J. Am. Chem. Soc. 1992, 114, 10508.
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