Unimolecular reaction, defined

The rate of formation of groups requires a modification of Eq. (68), since it is no longer possible to include breakage or exchange reactions. Adding the unimolecular reactions defined above, the new general rate equation becomes Eq. (124). 

From this expression, it is obvious that the rate is proportional to the concentration of A, and k is the proportionality constant, or rate constant, k has the units of (time) usually sec is a function of [A] to the first power, or, in the terminology of kinetics, v is first-order with respect to A. For an elementary reaction, the order for any reactant is given by its exponent in the rate equation. The number of molecules that must simultaneously interact is defined as the molecularity of the reaction. Thus, the simple elementary reaction of A P is a first-order reaction. Figure 14.4 portrays the course of a first-order reaction as a function of time. The rate of decay of a radioactive isotope, like or is a first-order reaction, as is an intramolecular rearrangement, such as A P. Both are unimolecular reactions (the molecularity equals 1). 

Figure 6. Diagram showing how the winding number n of the Feynman paths should be defined with respect to the cut line. In (a), the cut line (chains) is placed between (() = — and 2n — in (b), between (() = ti/4 and —In/A. In (c), the wave function describes a unimolecular reaction, in which the initial state occupies the (gray shaded) area shown. Feynman paths originate from all points within this area (inset) their winding number n is defined with respect to the common cut line.

On a phenomenological level, transitions between long-lived stable states can be described in terms of reaction rate constants. Consider, for instance a solution of two well-defined chemical species s3 and 38 that can interconvert through the unimolecular reaction... 

The experiments show that the dilution of all the monomers leads to a change of rate, and I contend that at the earliest stage of dilution the polymerizations are still mainly unimolecular and I offer an explanation for the effects of solvents on the rate of the unimolecular reactions. Since the rate constants, k, are defined by (4.1) and (4.14) they can only be calculated if [P+ M] is known. As explained in Section 3b, there are reasons for believing that for cyclopentadiene and for isobutene [P+ M = c, but for the former there are no results for solutions, and for the latter no c values are available, so that for these monomers could only be calculated for the bulk polymerizations. 

For this QRRK analysis we will define the zero of energy as the ground-state energy of the stabilized C molecule. As in QRRK the analysis of unimolecular reactions, assume that the excited C molecule consists of s identical oscillators, each with vibrational frequency v. When we write C ( ), this indicates that the excited intermediate species has been formed with n quanta of vibrational energy thus, its total energy is E = nhv above the ground-state energy of C (which we have arbitrarily set to zero). 

R. A. Marcus It certainly is a good point that transition state theory, and hence RRKM, provides an upper bound to the reactive flux (apart from nuclear tunneling) as Wigner has noted. Steve Klippenstein [1] in recent papers has explored the question of the best reaction coordinate, e.g., in the case of a unimolecular reaction ABC — AB + C, where A, B, C can be any combination of atoms and groups, whether the BC distance is the best choice for defining the transition state, or the distance between C and the center of mass of AB, or some other combination. The best combination is the one which yields the minimum flux. In recent articles Steve Klippenstein has provided a method of determining the best (in coordinate space) transition state [1]. 

Diels-Alder reaction, for example, is much better suited as a unimolecular reaction than the bimolecular cycloaddition because the former allows better control of precursors, in which the structural properties are well defined (96JA8755). Fragmentation of a molecule may be initiated by various methods depending on how the required energy is supplied. 

Most of the studies of ions formed by charge transfer have been concentrated on the unimolecular reactions of M+ ions formed in well-defined internal-energy states (e.g., fragmentation patterns6) and more recently have been concerned with rate-coefficient measurements.118 Some work has also been reported on consecutive ion-molecule reactions of M+ ions produced in well-defined internal states (mostly... 

A large number of radical reactions proceed by redox mechanisms. These all require electron transfer (ET), often termed single electron transfer (SET), between two species and electrochemical methods are very useful to determine details of the reactions (see Chapter 6). We shall consider two examples here - reduction with samarium di-iodide (Sml2) and SRN1 (substitution, radical-nucleophilic, unimolecular) reactions. The SET steps can proceed by inner-sphere or outer-sphere mechanisms as defined in Marcus theory [19,20]. 

Elements of classical dynamics of unimolecular reactions in particular, the Slater theory for indirect reactions, where the molecule is modeled as a set of uncoupled harmonic oscillators. Reaction is defined to occur when a particular bond length attains a critical value, and the rate constant is given as the frequency with which this occurs. 

Recently, two basic questions of chemical dynamics have attracted much attention first, is it possible to detect ( film ) the nuclear dynamics directly on the femtosecond time scale and second, is it possible to direct (control) the nuclear dynamics directly as it unfolds These efforts of real-time detection and control of molecular dynamics are also known as femtosecond chemistry. Most of the work on the detection and control of chemical dynamics has focused on unimolecular reactions where the internuclear distances of the initial state are well defined within, of course, the quantum mechanical uncertainty of the initial vibrational state. The discussion in the following builds on Section 7.2.2, and we will in particular focus on the real-time control of chemical dynamics. It should be emphasized that the general concepts discussed in the present section are not limited to reactions in the gas phase. 

In Sect. 7, we raised the question of what were the chemical stimuli to which the reactivity indices defined in Sect. 6, the softness kernels, were presumed to be the responses, our seventh issue. Now there are various broad categories of reactions to be considered, unimolecular, bimolecular, and multimolecular. The former occur via thermal activation over a barrier, tunneling through the barrier, or some combination of both. There is no stimulus, and the softness kernels defined as responses of the electron density to changes in external or nuclear potential are irrelevant. For the study of unimolecular reactions, one needs only information about the total energy in the relevant configuration space of the molecule. 

A chain reaction is defined as a reaction in which the products react with the initial material, and, again, the products of this second step then react with more of the initial material, thus involving a series of reaction cycles. A series of reactions can be built up in such a way as to give an equilibrium or steady state in which the fconcentration of material for the rate-determining step is directly proportional to the concentration of the initial material. These are the conditions required for a unimolecular reaction. Thus we have another way of explaining the independence of reaction rates and total collision frequency. 

There are a number of open issues associated with statistical descriptions of unimolecular reactions, particularly in many-dimensional systems. One fundamental issue is to find a qualitative criterion for predicting if a reaction in a many-dimensional system is statistical or nonstatistic al. In a recent review article, Toda [17] discussed different aspects of the Arnold web — that is, the network of nonlinear resonances in many-dimensional systems. Toda pointed out the importance of analyzing the qualitative features of the Arnold web— for example, how different resonance zones intersect and how the intersections further overlap with one another. However, as pointed out earlier, even in the case of fully developed global chaos it remains challenging to define a nonlocal reaction separatrix and to calculate the flux crossing the separatrix in a manydimensional phase-space. 

Equation (61) is ambiguous until we state at precisely what temperature, pressure, and composition the reaction-rate function co is to be evaluated. The ambient pressure p and the initial mass fractions Y- are reasonable first approximations for the pressure and composition. Since the principal profile changes (for example, the development of the local temperature maximum) occur well within the hot inert stream throughout most of the combustion development region, the most reasonable temperature to use in CO in equation (61) is T2. Thus, the specification co = co(T2, p, YJ J serves to define the right-hand side of equation (61) unambiguously. For example, for the unimolecular reaction F -> P, it can be seen from equations (1-8) and (1-43) that equation (61) then becomes... 

Define the following expressions empirical method, metastable equilibrium, kindling temperature, thermostat, interface, dynamic steady state, unimolecular reaction, bimolecular reaction, homogeneous reaction, heterogeneous reaction. 

This is similar to Eqn. 7.24, the double reciprocal equation for the simple unimolecular reaction. The difference is that in Eqn. 7.26, K]vi is modified and should be considered to be K vi(apparent) which is defined in Eqn. 7.27. 

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