J. Phys. Chem. 1996, 100, 8003-8007
8003
Unimolecular Reaction Dynamics from Kinetic Energy Release Distributions. 2. A Study of the Reaction C6H5Br+ f C6H5+ + Br by the Maximum Entropy Method P. Urbain,† F. Remacle,‡ B. Leyh,‡ and J. C. Lorquet* De´ partement de Chimie, Sart Tilman, B6, UniVersite´ de Lie` ge, B4000 Lie` ge 1, Belgium ReceiVed: October 17, 1995; In Final Form: December 18, 1995X
The kinetic translational energy released in the unimolecular fragmentation reaction C6H5Br+ f C6H5+ + Br has been experimentally studied in the microsecond time scale and theoretically analyzed by the maximum entropy formalism. The appropriate functional form relating the actual distribution to its prior distribution (eq 2.3) involves the square root of the kinetic energy (i.e. the momentum associated with the relative translational energy). A value of 0.26 ( 0.02 is obtained for the entropy deficiency distribution at an internal energy of 0.85 eV above the reaction threshold. From this value, it can be concluded that about 77% of the transition state phase space is effectively sampled.
I. Introduction The way in which the internal energy in excess of the minimal energy required for decomposition is partitioned among the translational, rotational, and vibrational degrees of freedom of the products is important information in studies of molecular reaction dynamics. Many different theoretical models have been developed, based either on the RRKM theory in its elementary version1 or on an extension of it,2,3 on phase space theory,4,5 on the definition of an effective temperature,6,7 or on a surprisal analysis based on the maximum entropy method.8-11 The present paper is devoted to a study by the latter method of the translational energy released in unimolecular dissociation processes observed in an ion beam. For neutrals, the role of translational energy has been studied in bimolecular collisions, and the informative constraints have been identified.8-14 In the case of ionized molecules, translational kinetic energy release distributions have been used to characterize various chemical processes.15-20 However, the maximum entropy method in the form of a surprisal analysis has been relatively seldom used17,21-23 despite the fact that this method has amply demonstrated its usefulness in the analysis of vibrational and rotational energy disposal.8-14,24 This results from the fact that, with respect to other experimental techniques (e.g., those involving the use of laser light), mass spectrometric experimentation presents several features of its own. We will restrict ourselves to the discussion of kinetic energy release distributions associated with the socalled metastable dissociations. (i) The experiment is carried out at very low pressures. Bimolecular collisions can be either suppressed or brought under control. (ii) The translational energy of a charged particle is much more readily measured than that of a neutral fragment. (iii) The experiment is time-selected. Only those dissociation processes that take place on a given time scale (usually referred to as the “metastable window”, i.e., on the order of a microsecond) are registered. This implies a corresponding selection on the internal energy. However, the selection is not sharp. It is characterized by a certain collection efficiency function T(E), which presents a nearly symmetrical bell shape, * Corresponding author. Fax: 32-41-662933. E-mail: u211101@ vm1.ulg.ac.be. † Boursier FRIA, Belgium. ‡ Chercheur Qualifie ´ , FNRS, Belgium. X Abstract published in AdVance ACS Abstracts, April 1, 1996.
S0022-3654(95)03069-3 CCC: $12.00
centered on a particular internal energy Es. The experimentally measured kinetic energy distribution involves an average over the function T(E). (iv) Interestingly enough, this bell-shaped function can be shifted along the energy axis by modifying the time selection in two ways. The value of Es (i.e., the most represented internal energy) can be modified by varying the extraction potentials applied in the ion source. Alternatively, the analysis can be carried out in a multisector mass spectrometer presenting several field-free regions. Only the former method will be used in the present paper. The latter is left for a further study. As a first application, we have chosen the loss of bromine from ionized bromobenzene. Malinovich et al.25 concluded that this process is a simple bond cleavage characterized by a totally loose transition state. The variation of the rate constant with internal energy has also been determined in a relatively broad range by different techniques.25-31 II. The Maximum Entropy Method Let P(|E) denote the probability of releasing a translational energy from a molecular ion having an internal energy E. The purpose of a maximum entropy analysis is to identify the constraints that preclude this distribution from being fully statistical and, in a second step, to elucidate the physical significance of these constraints. The analysis starts with the prior distribution, denoted P0(|E), which represents the most statistical situation, i.e., a state of affairs where populations are determined by energy level densities alone. If denotes the released translational energy and E the total internal energy of the decaying molecule, then8-11,24
P0(|E) ) A(E)xN(E - )
(2.1)
where N(E - ) represents the vibrational-rotational energy level density of the pair of fragments and A(E) is a normalization factor. The choice of the prior distribution should not be interpreted as a rough theoretical estimate but as the least biased reference distribution.8-14 Alternative choices would require a knowledge of the intramolecular forces acting on the separating fragments up to the range of internuclear distances where the rotational barriers are found, and this information is not presently available. © 1996 American Chemical Society
8004 J. Phys. Chem., Vol. 100, No. 19, 1996
Urbain et al.
In the maximum entropy formalism, the actual distribution P(|E) is related to the prior distribution in the following way: n
P(|E) ) P0(|E) exp(-λ0 - ∑λrAr)
(2.2)
r)1
where the quantities Ar are observables and the λr are Lagrange parameters. In the spirit of the method, the number of terms occurring in eq 2.2 should be as small as possible. The quantities Ar are then called informatiVe obserVables or constraints. For processes that are generated by a nonselective excitation, as in the present case, the input function is usually a broad asymmetric peak, and a single constraint is often sufficient to account for all of the major features. The identification of the relevant constraints provides information on the dynamics underlying a chemical reaction. The first (and possibly also the second) moments of the distribution are often good candidates. In principle, however, the appropriate functional form of the constraints is dictated by the physical mechanism which controls the final state distribution, and it is sometimes appropriate to use in eq 2.2 a slightly more complicated expression involving, for example, nonintegral powers of the observable. For example, in the case of bimolecular collisions, it has been found by Levine et al.12-14 that the proper informative observable was not the kinetic energy , but rather its square root, i.e., the momentum of the colliding particle. In this particular case, the appropriate functional dependence can be derived from the Born approximation when applied to the transfer of a light atom upon collision. For the unimolecular dissociation of the ion C6H5Br+ investigated here, it can be argued that such a reaction is essentially a vibrational predissociation. The rate of decay of this type of process has been extensively studied in the particular case of van der Waals complexes. Well above the dissociation threshold, the decay widths were found to obey the so-called “exponential momentum gap law”.32,33 Since the asymptotic yield in each individual decay channel is in a first approximation mainly governed by its quantum mechanical width, the final state kinetic energy distribution P(|E) is expected to exhibit an exponential dependence with respect to the momentum of the released particle, i.e., to the square root of the translational energy. The appropriate form of eq 2.2 is thus in this case
P(|E) ) P0(|E) exp[-λ0 - λ11/2]
(2.3)
As discussed in section IV, this equation is found to give a much better fit with experiment than those where the kinetic energy appears at the first or second power and is even better than when the first two moments are both used as constraints. However, the comparison between eq 2.3 and experiment cannot be done right away, since the experimental data involve an average over a range of internal energies. The measured distribution of translational energies P ˜ () is given by
P ˜ () ) ∫ T(E) P(|E) dE E
(2.4)
where T(E) is the collection efficiency corresponding to the metastable dissociation under study. This function, which is displayed in Figure 1, will be discussed in more detail in section IV. III. Experiment The experimental kinetic energy release distribution for the C6H5Br+ f C6H5+ + Br reaction has been extracted from ion kinetic energy spectra obtained under different experimental
Figure 1. Collection efficiency T(E) (eq 4.1) for the metastable dissociation C6H5Br+ f C6H5+ + Br taking place in the first field-free region of the AEI-MS9 spectrometer. The solid line corresponds to a fragment ion translational energy of 4 keV, whereas the dashed line corresponds to a fragment ion translational energy of 1 keV.
conditions. These spectra have been recorded for metastable dissociations taking place in the first field-free region of a forward-geometry AEI-MS9 mass spectrometer using two different scanning techniques: first, the accelerating voltage scan method34 and, second, the linked scanning of the electrostatic analyzer and of the magnet at a constant B2/E ratio.35 In the accelerating voltage scan technique, the kinetic energy of the fragment ions (C6H5+ here) is kept constant, while the accelerating voltage is scanned to detect the dissociation events leading to this fragment ion with a given distribution of kinetic energy releases. Both techniques provide us with an ion kinetic energy spectrum, which is in fact a parent ion spectrum and which contains the information concerning the kinetic energy released on the fragments. The ion kinetic energy is, however, measured in the apparatus reference frame (we shall usually call this energy the translational energy), whereas from a physical point of view, one is interested in the kinetic energy released in the center-of-mass frame. This latter information can be obtained by numerical differentiation of the measured ion kinetic energy spectrum followed by a transformation of variables from the laboratory coordinates to the center-of-mass coordinates.15,20,36 This differentiation procedure is rigorously valid only when instrumental broadening and y and z discrimination effects can be neglected. In this case, the basis functions, which are equal to the ion kinetic energy spectrum corresponding to a kinetic energy release distribution represented by a Dirac delta function, are rectangular pulse functions. When instrumental effects perturb the spectra, these basis functions are no longer so simple: they can be obtained either numerically37-39 or analytically,40,41 with a few reasonable assumptions. Starting from the calculated basis functions, a fitting algorithm leads to the kinetic energy release distribution. In the present case, however, the kinetic energy released on the C6H5+ fragment (45 eV in the laboratory frame for a fragment ion translational energy equal to 4 keV) is low enough for us to neglect the discrimination effects. On the other hand, the resolving power (vide infra) is sufficient to make a deconvolution procedure unnecessary. We are nevertheless aware that the study of reactions where larger kinetic energies are released will require the use of more elaborate methods37-41 to extract the kinetic energy release distribution. The differentiation procedure adopted in this work has been carried out using the Holmes-Osborne method.15 This consists in fitting the experimental ion kinetic energy spectrum to the product of a Gaussian function and a third-order polynomial. This expansion is then differentiated analytically to get the kinetic energy release distribution P ˜ (). Both scanning tech-
Study of the Reaction C6H5Br+ f C6H5+ + Br
J. Phys. Chem., Vol. 100, No. 19, 1996 8005 A. Collection Efficiency. Let us first discuss the collection efficiency function T(E) which appears in eq 2.4. It is a bellshaped function which depends on the ion lifetimes sampled in the experiment and on the function k(E) relating the rate constant to the internal energy E. The sampled lifetimes depend in turn on the electric fields applied in the ion source. We use for T(E) the following normalized analytical expression:
T(E) ) B[exp(-k(E)τ1) - exp(-k(E)τ2)]
(4.1)
The times τ1 and τ2 are entry and exit times in the field-free region, discussed in section III, and B is the normalization constant. The function k(E) has been experimentally determined by various authors (under the assumption that the decay is exponential; this assumption is also implicit in eq 4.1). Its dependence on the internal energy E is fitted to the following empirical form using the different experimental values available from the literature:25,27-31
k(E) ) kopt(E/Es)ν
Figure 2. (top) Ion kinetic energy spectrum measured for the metastable dissociation C6H5Br+ f C6H5+ (4 keV) + Br. The two peaks derive from the two isotopes of the bromine atom. This spectrum has been recorded using the accelerating voltage scan method. (bottom) Kinetic energy release distribution P ˜ () (eq 2.4). The error bars are deduced from a statistical analysis made on 18 measurements.
niques have been shown to lead to identical results within experimental error. A typical parent ion spectrum together with the corresponding kinetic energy distribution is shown in Figure 2. The ion-focusing conditions have been optimized to reach the best possible resolution and to avoid distortion effects which manifest themselves as unsymmetrical ion signals. The exit slit of the electrostatic analyzer has been closed to reach a kinetic energy resolution of ∆E/E ) 10-3. The experimental broadening is therefore much smaller than the broadening induced by the release of kinetic energy. Accelerating voltage scan spectra have been recorded for fragment ion translational energies equal to 1 and 4 keV, respectively. For the B2/E spectra, the parent ion translational energy was equal to 8 keV. The other experimental conditions were standard ones: trap current, 30 µA; electron energy, 70 eV; ion source pressure (measured at an ion gauge located approximately 15 cm from the ionization chamber), 10-4 Pa. Bromobenzene provided by Fluka (99.5% stated purity) was used without further purification. As will be shown in the next section, a key piece of information for the calculation of the transmission function T(E) is the range of ion lifetimes (τ1, τ2) sampled in the experiment. The entry time in the first field-free region, τ1, is the sum of the residence time of the ions in the ionization chamber and of the flight-time from this chamber to the source exit slit. These times depend in turn mainly on the accelerating voltage, on the repeller voltage, and on the distances between the relevant optical elements and on the ion mass. All these parameters are known, allowing the calculation of τ1 and τ2 by elementary physics. For the 4 keV experiments, τ1 and τ2 are equal to 1.74 and 3.77 µs, respectively. The corresponding values at 1 keV are 4.16 and 8.22 µs. IV. Fitting Procedure The experimentally observed kinetic energy distributions are fitted to the maximum entropy functional form given by eqs 2.1, 2.3, and 2.4.
(4.2)
where kopt is the value of the rate constant at the energy Es at which T(E) reaches its maximum. The optimal value of the parameter ν is equal to 6.00 ( 0.05. In the present experiments, for a fragment ion translational energy of 4 keV, the collection efficiency reaches a maximum for an internal energy of the ion C6H5Br+ (measured in excess to the dissociation asymptote) equal to 0.85 eV. The corresponding rate constant is then equal to 3.8 × 105 s-1. When the fragment ion translational energy is equal to 1 keV, these values become 0.75 eV and 1.7 × 105 s-1, respectively. This procedure leads to the functions presented in Figure 1. B. Fitting Function. From a physical point of view, the distribution P(|E) in eq 2.3 must be normalized. This condition ensures that the averaged distribution P ˜ () is automatically normalized. Substituting eqs 2.1 and 2.3 into eq 2.4, one obtains the following expression:
P ˜ () ) ∫ T(E)P0(|E) exp(-λ0 - λ11/2) dE ∞
T(E)1/2N(E - ) exp(-λ11/2)
) ∫ dE ∞
C(E)
(4.3)
where C(E) is the normalization factor of P(|E).
C(E) ) exp(λ0)/A(E) ) ∫0 d 1/2 exp(-λ11/2)N(E - ) (4.4) E
C. Prior Distribution. The prior distribution P0(|E) has been generated by a Beyer-Swinehardt algorithm42 where the rotational parameters and the vibrational frequencies of the C6H5+ ion have been calculated ab initio at the HF 6-31 G* level.43 It is plotted in Figure 3 for three different values of the internal energy E that correspond to a significant intensity of the collecting efficiency T(E) (Figure 1) computed for a fragment ion translational energy of 4 keV. D. Comparison with Experiment. Figure 4 shows the result of a least-squares fit of eq 4.3 to the 4 keV experimental data under the assumption that the Lagrange parameter λ1 is independent of the internal energy E. A similar good fit has been obtained for the 1 keV data. Formally, the Lagrange parameter λ1 depends on the parent ion internal energy. In principle, this dependence could be investigated by changing the fragment ion translational energy from 4 keV to 1 keV. As can be seen from Figure 1, however, the induced changes are limited and the corresponding T(E)
8006 J. Phys. Chem., Vol. 100, No. 19, 1996
Urbain et al. The prior entropy is always larger than or equal to S, and the difference is called the entropy deficiency DS. Taking into account eqs 2.3, 5.1, and 5.2, one has8-11
DS ) Sprior - S ) -λ0 - λ1〈1/2〉
(5.3)
〈1/2〉 ) ∫0 1/2 P(|E) d
(5.4)
where E
Figure 3. Prior kinetic energy distributions P0(|E) (eq 2.1) calculated for E ) 0.7 eV, 0.85 eV, and 1.0 eV above the C6H5+ + Br dissociation asymptote.
Figure 4. Experimental (eq 2.4) and fitted (eq 4.3) kinetic energy release distributions obtained for the dissociation C6H5Br+ f C6H5+ (4 keV) + Br.
functions do not span significantly different energy ranges. Values of 6.2 and 6.7 were obtained for λ1 at 1 and 4 keV, respectively. This variation is too small with respect to the experimental errors to lead to a definite conclusion in the limited energy range available. As a test of internal consistency, we studied the influence of the internal energy distribution T(E) on the average kinetic energy release, 〈〉, defined by
˜ () d 〈〉 ) ∫0 P ∞
(4.5)
As shown by eq 4.3, 〈〉 depends on λ1 and on P0(|E). Both experimental and numerical results indicate that this variation is negligible when one switches from 4 keV to 1 keV, in agreement with independent data obtained by Burgers and Holmes on the same ionic system.31 Altogether, the quality of the fits obtained at 1 and 4 keV supports the present assumptions: (i) the functional dependence involving the square root of appearing in eq 2.3 and (ii) the independence of λ1 with respect to the internal energy E. V. Discussion How can the shape of the functions P ˜ () (eq 4.3) and P(|E) (eq 2.3) be related to the statistical character of the reaction? By definition, the prior distribution P0(|E) represents the most statistical situation, that of maximal entropy. At a given energy E, the actual distribution P(|E) is characterized by an entropy S:
S ) -∫0 P(|E) ln[P(|E)] d E
(5.1)
while the entropy Sprior is defined for the prior distribution P0(|E) by
S
prior
) -∫
E 0 P (|E) 0
0
ln[P (|E)] d
(5.2)
A nonzero value for DS amounts to saying that the phase space sampled by the pair of fragments is reduced with respect to its maximal value by an amount equal to exp(-DS). However, most chemists like to describe a unimolecular reaction in the framework of a transition state model. As a result, they would like to interpret any deviation between the prior and the actual distributions (i.e., any nonzero value of the Lagrange parameters) in terms of a breakdown of two assumptions of the RRKM model.3,44,45 Within this model, a nonzero value of the surprisal parameter λ1 and of the associated entropy deficiency can result from two circumstances: (1) a possible incomplete randomization of the internal energy, in other words, an incomplete sampling of that part of phase space associated with the transition state; (2) the existence of interactions between the fragments as they separate after the transition state region, i.e., after having crossed the dividing surface in phase space, that brings about conversion of relative translational energy into rotational and vibrational excitation of the receding fragments.3,46,47 The latter process probably plays an insignificant role in the present case for several reasons. (a) The average kinetic energy release defined in eq 4.5 is fairly small (〈〉 ≈ 0.06 eV ) 480 cm-1). (b) Bobsleigh effects and rotational excitation are not expected on geometrical grounds: the reaction presumably consists of a simple bond cleavage involving the relative motion of a structureless atom in the plane of a rigid ring. (c) According to previous studies,25 the reaction can be described as a simple bond cleavage via a totally loose transition state. From our experimental results at 1 and 4 keV, we obtain at an internal energy of E ) 0.85 eV a value DS equal to 0.26 ( 0.03. The corresponding reduction factor exp(-DS) is equal to 0.77 ( 0.02. Therefore, if the final state interactions between the separated fragments are assumed to be negligible, then it can be deduced that approximately 77% of that fraction of phase space offered to the transition state is effectively sampled. This fraction should be sufficiently large to ensure the applicability of a statistical picture of the reaction as postulated in the RRKM model, and experimentally confirmed.25 Better separated time windows and associated T(E) functions could be obtained by working in different field-free regions of a multisector mass spectrometer. It should then be possible to determine the variation of the Lagrange parameters and of the associated entropy deficiency with E. In other words, a worthwhile goal would be to study the sampling of phase space as a function of the internal energy. This question will be considered in future work. Acknowledgment. We wish to thank Dr. A. J. Lorquet for the ab initio calculation of the vibrational frequencies of the C6H5+ ion. This work has been supported by a research grant from the Fonds de la Recherche Fondamentale Collective (Belgium). References and Notes (1) Forst, W. Theory of Unimolecular Reactions; Academic Press: New York, 1973.
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