Unimolecular dissociation RRKM theory

A situation that arises from the intramolecular dynamics of A and completely distinct from apparent non-RRKM behaviour is intrinsic non-RRKM behaviour [9], By this, it is meant that A has a non-random P(t) even if the internal vibrational states of A are prepared randomly. This situation arises when transitions between individual molecular vibrational/rotational states are slower than transitions leading to products. As a result, the vibrational states do not have equal dissociation probabilities. In tenns of classical phase space dynamics, slow transitions between the states occur when the reactant phase space is metrically decomposable [13,14] on the timescale of the imimolecular reaction and there is at least one bottleneck [9] in the molecular phase space other than the one defining the transition state. An intrinsic non-RRKM molecule decays non-exponentially with a time-dependent unimolecular rate constant or exponentially with a rate constant different from that of RRKM theory. 

In the above discussion it was assumed that the barriers are low for transitions between the different confonnations of the fluxional molecule, as depicted in figure A3.12.5 and therefore the transitions occur on a timescale much shorter than the RRKM lifetime. This is the rapid IVR assumption of RRKM theory discussed in section A3.12.2. Accordingly, an initial microcanonical ensemble over all the confonnations decays exponentially. However, for some fluxional molecules, transitions between the different confonnations may be slower than the RRKM rate, giving rise to bottlenecks in the unimolecular dissociation [4, ]. The ensuing lifetime distribution, equation (A3.12.7), will be non-exponential, as is the case for intrinsic non-RRKM dynamics, for an mitial microcanonical ensemble of molecular states. 

Stumpf M, Dobbyn A J, Keller H-M, Hase W L and Schinke R 1995 Quantum mechanical study of the unimolecular dissociation of HO2 a rigorous test of RRKM theory J. Chem. Phys. 102 5867-70... 

The quasi-equilibrium theory (QET) of mass spectra is a theoretical approach to describe the unimolecular decompositions of ions and hence their mass spectra. [12-14,14] QET has been developed as an adaptation of Rice-Ramsperger-Marcus-Kassel (RRKM) theory to fit the conditions of mass spectrometry and it represents a landmark in the theory of mass spectra. [11] In the mass spectrometer almost all processes occur under high vacuum conditions, i.e., in the highly diluted gas phase, and one has to become aware of the differences to chemical reactions in the condensed phase as they are usually carried out in the laboratory. [15,16] Consequently, bimolecular reactions are rare and the chemistry in a mass spectrometer is rather the chemistry of isolated ions in the gas phase. Isolated ions are not in thermal equilibrium with their surroundings as assumed by RRKM theory. Instead, to be isolated in the gas phase means for an ion that it may only internally redistribute energy and that it may only undergo unimolecular reactions such as isomerization or dissociation. This is why the theory of unimolecular reactions plays an important role in mass spectrometry. 

B. Unimolecular Dissociation Rates RRKM Theory and Distribution of Resonances... 

The RRKM theory of unimolecular reactions predicts that the rate constant for dissociation will be given by eq. (5-3). The probability of populating a state with energy Ev restricted into the chromophore vibrations is proportional to the ratio of the density of van der Waals states at E — Ev to that at ... 

The most common theory used to treat bimolecular recombination and unimolecular dissociation reactions in the literature is a statistical theory, "RRKM" theory (Rice, Ramsperger, Kassel, Marcus) [34-39]. However, symmetrical iso-topomers such as and have fewer intramolecu-... 

The focus of this chapter is a review of the methodologies employed in a priori implementations of RRKM theory for the collisionless dissociation/ isomerization steps in gas-phase unimolecular reactions. Special attention will be paid to recent developments, particularly those that have proven their utility through substantive applications. With microscopic reversibility, RRKM treatments of the dissociation process are directly applicable to the reverse bimolecular associations. Furthermore, some of the more interesting illustrations are for bimolecular reactions and so we do not limit our discussion of RRKM theory to unimolecular reactions. However, one should bear in mind that TST was originally derived for bimolecular reactions and the specific term RRKM theory is really only applicable to the unimolecular direction. 

Several conclusions can be drawn from Eqs. (76) and (77). First, the influence of fluctuations is the largest when the number of open channels u is of the order of unity, because then the distribution Q k) is the broadest. Second, the effect of a broad distribution of widths is to decrease the observed pressure dependent rate constant as compared to the delta function-like distribution, assumed by statistical theories [288]. The reason is that broad distributions favor small decay rates and the overall dissociation slows down. This trend, pronounced in the fall-of region, was clearly seen in a recent study of thermal rate constants in the unimolecular dissociation of HOCl [399]. The extremely broad distribution of resonances in HOCl caused a decrease by a factor of two in the pressure-dependent rate, as compared to the RRKM predictions. The best chances to see the influence of the quantum mechanical fluctuations on unimolecular rate constants certainly have studies performed close to the dissociation threshold, i.e. at low collision temperatures, because there the distribution of rates is the broadest. 

Modern unimolecular theory has its origins in the work of Rice, Ramsberger and Kassel [44] who investigated the rate of dissociation of a molecule as a function of energy. Marcus and Rice [44] subsequently extended the theory to take account of quantum mechanical features. This extended theory, referred to as RRKM theory, is currently the most widely used approach and is usually the point of departure for more sophisticated treatments of unimolecular reactions. The key result of RRKM theory is that the microcanonical rate coefficient can be expressed as... 

State specific experiments can now test unimolecular rate theories by probing microcanonical rate coefficients. Moore and coworkers [45] have studied the dissociation of ketene close to the reaction threshold in an attempt to test RRKM theory. 

Experimental studies have had an enormous impact on the development of unimolecular rate theory. A set of classical thermal unimolecular dissociation reactions by Rabinovitch and co-workers [6-10], and chemical activation experiments by Rabinovitch and others [11-14], illustrated that the separability and symmetry of normal modes assumed by Slater theory is inconsistent with experiments. Eor many molecules and experimental conditions, RRKM theory is a substantially more accurate model for the unimolecular rate constant. Chemical activation experiments at high pressures [15,16] also provided information regarding the rate of vibrational energy flow within molecules. Experiments [17,18] for which molecules are vibrationally excited by overtone excitation of a local mode (e.g. C-H or O-H bond) gave results consistent with the chemical activation experiments and in overall good agreement with RRKM theory [19]. 

Variational RRKM theory is particularly important for unimolecular dissociation reactions, in which vibrational modes of the reactant molecule become translations and rotations in the products [22]. For CH CH3-1-H dissociation there are three vibrational modes of this type, i.e. the C—H stretch which is the reaction coordinate and the two degenerate H—CH bends, which first transform from high-frequency to low-frequency vibrations and then hindered rotors as the H—C bond ruptures. These latter two degrees of freedom are called transitional modes [24,25]. C2Hg 2CH2 dissociation has five transitional modes, i.e. two pairs of degenerate CH rocking/rotational motions and the CH torsion. 

The first classical trajectory study of unimolecular decomposition and intramolecular motion for realistic anharmonic molecular Hamiltonians was performed by Bunker [12,13]. Both intrinsic RRKM and non-RRKM dynamics was observed in these studies. Since this pioneering work, there have been numerous additional studies [9,17,30,M,65,66 and 62] from which two distinct types of intramolecular motion, chaotic and quasiperiodic [14], have been identified. Both are depicted in figure A3.12.7. Chaotic vibrational motion is not regular as predicted by the normal-mode model and, instead, there is energy transfer between the modes. If all the modes of the molecule participate in the chaotic motion and energy flow is sufficiently rapid, an initial microcanonical ensemble is maintained as the molecule dissociates and RRKM behaviour is observed [9]. For non-random excitation initial apparent non-RRKM behaviour is observed, but at longer times a microcanonical ensemble of states is formed and the probability of decomposition becomes that of RRKM theory. 

RRKM theory is a microcanonical transition state theory and as such, it gives the connection between statistical unimolecular rate theory and the transition state theory of thermal chemical reaction rates. Isomerization or dissociation of an energized molecule A is assumed in RRKM theory to occur via the mechanism... 

See, for example, D. L. Bunker, /. Chem. Phys., 40,1946 (1963). Monte Carlo Calculations. IV. Further Studies of Unimolecular Dissociation. D. L. Bunker and M. Pattengill,/. Chem. Phys., 48, 772 (1968). Monte Carlo Calculations. VI. A Re-evaluation erf Ae RRKM Theory of Unimolecular Reaction Rates. W. J. Hase and R. J. Wolf, /. Chem. Phys., 75,3809 (1981). Trajectory Studies of Model HCCH H -P HCC Dissociation. 11. Angular Momenta and Energy Partitioning and the Relation to Non-RRKM Dynamics. D. W. Chandler, W. E. Farneth, and R. N. Zare, J. Chem. Phys., 77, 4447 (1982). A Search for Mode-Selective Chemistry The Unimolecular Dissociation of t-Butyl Hydroperoxide Induced by Vibrational Overtone Excitation. J. A. Syage, P. M. Felker, and A. H. Zewail, /. Chem. Phys., 81, 2233 (1984). Picosecond Dynamics and Photoisomerization of Stilbene in Supersonic Beams. II. Reaction Rates and Potential Energy Surface. D. B. Borchardt and S. H. Bauer, /. Chem. Phys., 85, 4980 (1986). Intramolecular Conversions Over Low Barriers. VII. The Aziridine Inversion—Intrinsically Non-RRKM. A. H. Zewail and R. B. Bernstein,... 

The subsequent step, unimolecular dissociation, has been explained fully in terms of RRKM and quasiequilibrium theory (QET) theories (see Section 6.7). 

Here, most quantities are defined above and k(e + Ei) = k(E ) is the unimolecular dissociation rate constant, evaluated using modern statistical theories, such as Rice-Ramsperger-Kassel-Marcus (RRKM) theory. Note that Equation (8) combines the distribution of deposited energies (Equation (5)) with the probability that the complex dissociates in time r (term in square brackets), and a summation over the internal energy available to the reactants. Importantly, the integration recovers Equation (2) when the dissociation rate, A ( ), is faster than the experimental time scale, such that the term in brackets is unity. 

The QET is formally identical to the Rice-Ramsperger-Kassel-Marcus (RRKM) theory of unimolecular decay, in which the rate constant for dissociation to reaction products of an energized species with total angular momentum J and internal energy E over a barrier of Eq is given by the following relation ... 

As shown in Figure 20.4, the nature of the spectrum, near the unimolecular threshold, indicates whether the unimolecular dynamics will be intrinsically RRKM or non-RRKM. RRKM theory assumes irregular chaotic dynamics and, if this is the nature of the spectrum near the dissociation threshold, intrinsic RRKM behavior is expected. In contrast, if an appreciable fraction of the spectrum is regular near the dissociation threshold, the dynamics should be intrinsically non-RRKM. 

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