RRKM theory unimolecular dissociation rates

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

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. 

Thus all points of the PES can be obtained (in principle) and unimolecular dissociation rate constants expressed as a function of internal energy. Nowadays, the vibrational frequencies can be calculated at a high level of theory, and the sum and density of states obtained by direct count without any particular problem for small or medium sized systems, and thus RRKM kinetics were employed to examine several unimolecular dissociations [150]. Even complex kinetic schemes were solved to obtain the rate constant for product formation through different reaction channels [156, 157]. 

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. 

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 ... 

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]. 

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 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 ... 

RRKM theory is the well-known and consolidated statistical theory for unimolecular dissociation. It was developed in the late 1920s by Rice and Ramsperger [141, 142] and Kassel [143], who treated a system as an assembly of s identical harmonic oscillators. One oscillator is truncated at the activation energy Eq. The theory disregards any quantum effect and the approximation of having all identical is too cmde, such that the derived equation for micro canonical rate constant, k(E),... 

Figure 7 The dissociation rate constants of C6H6 + ions as a function of the ion internal energy. The solid lines are calculated rate constants using the statistical theory of unimolecular decay (RRKM). Reproduced with permission from Baer T, Willet GD, Smith D and Phillips JS (1979) The dissociation dynamics of internal energy selected CeHe. Journal of Chemical Physics 70 4076 085.

Figure 7.14 Steps in the measured and (RRKM) computed reaction rate for the unimolecular dissociation. Left the dissociation of CD2CO to triplet CD2, as a function of the photolysis energy. This channel has a barrier, see Figure 7.13 [adapted from S. K. Kim, E. R. Lovejoy, and C. B. Moore, J. Chem. Phys. 102, 3202 (1995) see also Lovejoy etal. (1992), Marcus (1992)). Right a similar plot for the dissociation of CH2CO to singlet CH2 in the very threshold region (adapted from Moore and Smith (1996)]. The dissociation is energetically possible already at threshold. Just above the threshold the energies for the steps match the rotational levels of CO. This correspondence Is confirmed by a match to the phase-space theory computation shown as a smooth line. The fine structure in the steps can be identified with the low rotational states of the parent CH2CO.

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