Transition-state theory unimolecular decompositions

Miller W H, Hernandez R, Moore C B and Polik W F A 1990 Transition state theory-based statistical distribution of unimolecular decay rates with application to unimolecular decomposition of formaldehyde J. Chem. Phys. 93 5657-66... 

For a temperature of 1000 K, calculate the pre-exponential factor in the specific reaction rate constant for (a) any simple bimolecular reaction and (b) any simple unimolecular decomposition reaction following transition state theory. 

At high temperatures and low pressures, the unimolecular reactions of interest may not be at their high-pressure limits, and observed rates may become influenced by rates of energy transfer. Under these conditions, the rate constant for unimolecular decomposition becomes pressure- (density)-dependent, and the canonical transition state theory would no longer be applicable. We shall discuss energy transfer limitations in detail later. 

Both unimolecular and bimolecular reactions are common throughout chemistry and biochemistry. Binding of a hormone to a reactor is a bimolecular process as is a substrate binding to an enzyme. Radioactive decay is often used as an example of a unimolecular reaction. However, this is a nuclear reaction rather than a chemical reaction. Examples of chemical unimolecular reactions would include isomerizations, decompositions, and dis-associations. See also Chemical Kinetics Elementary Reaction Unimolecular Bimolecular Transition-State Theory Elementary Reaction... 

Merle JK, Hayes CJ, Zalyubovsky SJ, Glover BG, Miller TA, HadadCM. Theoretical determinations of the ambient conformational distribution and unimolecular decomposition of n-propylperoxy radical. J Phys Chem A. 2005 109 3637-3646. Vereecken L, Peeters J. The 1,5-shift in 1-butoxy a case study in the rigorous implementation of transition state theory for a multirotamer system. / Chem Phys. 2003 119 5159-5170. 

Jitariu LC, Jones LD, Robertson SH, PiUing MJ, Hillier IH. Thermal rate coefficients via variational transition state theory for the unimolecular decomposition/isometiza-tion of 1-pentyl radical ab initio and direct dynamics calculations. J Phys Chem A. 2003 107 8607-8617. 

Recent mechanistic discussions of unimolecular decompositions of organic ions have invoked ion—molecule complexes as reaction intermediates [102, 105, 361, 634]. The complexes are proposed to be bound by long-range ion—dipole forces and to be sufficiently long-lived to allow hydrogen rearrangements to occur. The question of lifetime aside, there is more than a close similarity between the proposed ion—dipole intermediate and the assumed loose or orbiting transition state of phase space theory. 

It has been mentioned that phase space theory, i.e. assuming a loose transition state, has been able to explain the translational energy releases in the decomposition of certain ion—molecule collision complexes [485] and in some unimolecular decompositions measured by PIPECO (see Sect. 8.2). There is a larger number of translational energy releases from PIPECO and a body of data as to translational energy releases in source reactions of positive ions formed by El [162, 310] (Sect. 8.3.1) with which the predictions of phase space theory are in poor agreement. The predicted energy releases are too low. 

It seems worthwhile to examine critically this transcription of the Slater method into the standard absolute reaction rate theory. In the simple unimolecular bond break, it does appear reasonable that the coordinate q between the tvfo atoms A and B must reach and go beyond a critical extension q0 in order that decomposition takes place. In Slater s calculations account is taken of the different energies involved in stretching q to q0. In regarding q as the mode of decomposition in the transition state method, one must, however, first look at the potential energy surface. The decomposition path involves passage over the lowest possible barrier between reactants and products. It does not seem reasonable to assume that this path necessarily only involves motion of the atoms A and B at the activated complex. Possibly, a more reasonable a priori formulation in a simple decomposition process would be to choose q as the coordinate which tears the two decomposition fragments apart. Such a coordinate would lead roughly to the relation... 

The random lifetime assumption is perhaps most easily tested by classical trajectory calculations (Bunker, 1962 1964 Bunker and Hase, 1973). Initial momenta and coordinates for the Hamiltonian of an excited molecule can be selected randomly, so that a microcanonical ensemble of states is selected. Solving Hamilton s equations of motion, Eq. (2.9), for an initial condition gives the time required for the system to reach the transition state. If the unimolecular dynamics of the molecule are in accord with RRKM theory, the decomposition probability of the molecule versus time, determined on the basis of many initial conditions, will be exponential with the RRKM rate constant. That is, the decay is proportional to exp[-k( )t]. The observation of such an exponential distribution of lifetimes has been identified as intrinsic RRKM behavior. If a microcanonical ensemble is not maintained during the unimolecular decomposition (i.e., IVR is slower than decomposition), the decomposition probability will be nonexponential, or exponential with a rate constant that differs from that predicted by RRKM theory. The implication of such trajectory studies to experiments and their relationship to quantum dynamics is discussed in detail in chapter 8. 

STATISTICAL THEORY OF UNIMOLECULAR DECOMPOSITION 6.5.1 Optically Active Reactants or Transition States... 

The statistical dissociation rate constant can be calculated from the point of view of the reverse reaction, namely the recombination of the products to form a complex. This approach, commonly referred to as phase space theory (PST) (Pechukas and Light, 1965 Pechukas et al., 1966 Nikitin, 1965 Klots, 1971, 1972) is limited to reactions with no reverse activation energy, that is, reactions with very loose transition states. PST assumes the decomposition of a molecule or collision complex is governed by the phase space available to each product under strict conservation of energy and angular momentum. The loose transition state limit assumes that the reaction potential energy surface is of no importance in determining the unimolecular rate constant. 

Intrinsic non-RRKM behavior occurs when an initial microcanonical ensemble decays nonexponentially or exponentially with a rate constant different from that of RRKM theory. The former occurs when there is a bottleneck (or bottlenecks) in the classical phase space so that transitions between different regions of phase space are less probable than that for crossing the transition state [fig. 8.9(e)]. Thus, a micro-canonical ensemble is not maintained during the unimolecular decomposition. A limiting case for intrinsic non-RRKM behavior occurs when the reactant molecule s phase space is metrically decomposable into two parts, for example, one part consisting of chaotic trajectories which can decompose and the other of quasiperiodic trajectories which are trapped in the reactant phase space (Hase et al., 1983). If the chaotic motion gives rise to a uniform distribution in the chaotic part of phase space, the unimolecular decay will be exponential with a rate constant k given by... 

Now, let us find a little flaw in the theory equation (2-86) predicts only first-order behavior for the unimolecular reaction, something we know in fact is not true at low pressures. The reason for this failure in TST is the assumption of universal equilibrium between reactants and the transition state complex. At low pressures the collisional deactivation process becomes very slow, since collisions are infrequent, and the rate of decomposition becomes large compared to deactivation. In such an event, equilibrium cannot be established nearly every molecule which is activated will decompose to product. However, the magnitude of the rate of decomposition of the transition complex is much larger than the decomposition of the activated molecule in the collision theory scheme, so one must resist the temptation to equate the two. Since the transition state complex represents a configuration of the reacting molecule on the way from reactants to products, the activated molecule must be a precursor of the transition state complex. 

Marcus and Rice [8] have formulated a theory of unimolecular decompositions which is based on a transition-state model. The physical assumptions of the theory are essentially the same as those of the RRK theory, but the reaction mechanism is postulated as... 

The procedure described above is for sampling a statistical microcanoni-cal distribution at the TS. However, if the unimolecular decomposition of the molecule is non-statistical and not in accord with RRKM theory, a statistical population of the TS s energy levels is not expected. For such a situation it may be impossible to identify how the TS levels are populated and, thus, simulate the experiment. One approach would be to vary the population of the TS levels until the direct dynamics and experimental product energy distributions agree. However, such an approach assumes the level of electronic structure theory used in the simulation is sufficiently accurate. With recent experimental advances in laser spectroscopy and dynamics,it may become possible to excite specific vibrational levels at the transition state. It would be straightforward to simulate such experiments. 

A more fundamental situation for which dynamical theories are important is when the activated molecule undergoing unimolecular decomposition has vibrational states with lifetimes longer than the statistical RRKM lifetime of the molecule. This will occur if transitions between two (or more) groups of states are less probable than those leading to products. As a result, the molecule will not have a random lifetime distribution even if its initial internal energy distribution is random. Molecules which behave in this manner are said to be "intrinsically" non-RRKM. Long-lived vibrational states have been detected experimentally in highly excited molecules, and it is of considerable interest to determine what... 

A combined experimental and density functional theory (DFT) study of the thermal decomposition of 2-methyl-l,3-dioxolane, 2,2-dimethyl-1,3-dioxolane, and cyclopen-tanone ethylene ketal, in the gas phase, has established that acetaldehyde and the corresponding ketone are formed by a unimolecular stepwise mechanism concerted nonsynchronous formation of a four-centred cyclic transition state is rate determining and leads to unstable intermediates that then decompose rapidly through a concerted cyclic six-centred transition state. ... 

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