Unimolecular reactions diffusion theory

The first of the theoretical chapters (Chapter 9) treats approaches to the calculation of thermal rate constants. The material is familiar—activated complex theory, RRKM theory of unimolecular reaction, Debye theory of diffusion-limited reaction—and emphasizes how much information can be correlated on the basis of quite limited models. In the final chapt, the dynamics of single-collision chemistry is analyzed within a highly simplified framework the model, based on classical mechanics, collinear collision geometries, and naive potential-energy surfaces, illuminates many of the features that account for chemical reactivity. 

The second part (sections H and I) is devoted to a detailed discussion of the dynamics of unimolecular reactions in the presence and the absence of a potential barrier. Section H presents a critical examination of the Kramers approach. It is stressed that the expressions of the reaction rates in the low-, intermediate-, and high-friction limits are subjected to restrictive conditions, namely, the high barrier case and the quasi-stationary regime. The dynamics related to one-dimensional diffusion in a bistable potential is analyzed, and the exactness of the time dependence of the reaction rate is emphasized. The essential results of the non-Markovian theory extending the Kramers conclusions are also discussed. The final section investigates in detail the time evolution of an unimolecular reaction in the absence of a potential barrier. The formal treatment makes evident a two-time-scale description of the dynamics. 

Shalashilin and Thompson [46-48] developed a method based on classical diffusion theory for calculating unimolecular reaction rates in the IVR-limited regime. This method, which they referred to as intramolecular dynamics diffusion theory (IDDT) requires the calculation of short-time ( fs) classical trajectories to determine the rate of energy transfer from the bath modes of the molecule to the reaction coordinate modes. This method, in conjunction with MCVTST, spans the full energy range from the statistical to the dynamical limits. It in essence provides a means of accurately... 

A classical diffusion theory model has been proposed to calculate the rate of IVR between the reaction coordinate and the remaining bath modes of the molecule [345]. Following work by Bunker [324], the unimolecular dynamics will be non-ergodic (intrinsically non-RRKM) if A rrkm fciVR. For such a situation, the unimolecular decomposition will be exponential and occur with a rate constant equal to /sivr- The rate of IVR is modeled by assuming a random force between the bath modes and the reaction coordinate. The model was used to successfully analyze the intrinsic non-RRKM dynamics for Si2He -> 2SiH3 dissociation [345]. 

Guo. Y.. Shalashilin. D. V.. Krouse. J. A. and Thompson. D. L.(1999) Intramolecular dynamics diffusion theory approach to complex unimolecular reactions, J. Chem. Phys. 110, 5521-5525. 

Guo, Y., Thompson, D. L. and Miller, W. (1999) Thermal and microcanonical rates of unimolecular reactions from an energy diffusion theory approach, J.Phys.Chem. A, 103, 10308-10311. 

Most of the ingredients of the model described above have been formerly postulated in treatments of unimolecular reactions. In particular, the model for the barrier dynamics is inherent in the usual TST for unimolecular reactions involving polyatomic molecules, while taking the total molecular energy Ej as the important dynamic variable in the well is the underlying assumption in theories that use a master or a diffusion equation for Ej- as their starting point. 

Section II provides a summary of Local Random Matrix Theory (LRMT) and its use in locating the quantum ergodicity transition, how this transition is approached, rates of energy transfer above the transition, and how we use this information to estimate rates of unimolecular reactions. As an illustration, we use LRMT to correct RRKM results for the rate of cyclohexane ring inversion in gas and liquid phases. Section III addresses thermal transport in clusters of water molecules and proteins. We present calculations of the coefficient of thermal conductivity and thermal diffusivity as a function of temperature for a cluster of glassy water and for the protein myoglobin. For the calculation of thermal transport coefficients in proteins, we build on and develop further the theory for thermal conduction in fractal objects of Alexander, Orbach, and coworkers [36,37] mentioned above. Part IV presents a summary. 

The Langmuir-Hinshelwood picture is essentially that of Fig. XVIII-14. If the process is unimolecular, the species meanders around on the surface until it receives the activation energy to go over to product(s), which then desorb. If the process is bimolecular, two species diffuse around until a reactive encounter occurs. The reaction will be diffusion controlled if it occurs on every encounter (see Ref. 211) the theory of surface diffusional encounters has been treated (see Ref. 212) the subject may also be approached by means of Monte Carlo/molecular dynamics techniques [213]. In the case of activated bimolecular reactions, however, there will in general be many encounters before the reactive one, and the rate law for the surface reaction is generally written by analogy to the mass action law for solutions. That is, for a bimolecular process, the rate is taken to be proportional to the product of the two surface concentrations. It is interesting, however, that essentially the same rate law is obtained if the adsorption is strictly localized and species react only if they happen to adsorb on adjacent sites (note Ref. 214). (The apparent rate law, that is, the rate law in terms of gas pressures, depends on the form of the adsorption isotherm, as discussed in the next section.)... 

Another moderately successful approach to the theory of diffusion in liquids is that developed by Eyring (E4) in connection with his theory of absolute reaction rates (P6, K6). This theory attempts to explain the transport phenomena on the basis of a simple model for the liquid state and the basic molecular process occurring. It is assumed in this theory that there is some unimolecular rate process in terms of which the transport processes can be described, and it is further assumed that in this process there is some configuration that can be identified as the activated state. Then the Eyring theory of reaction rates is applied to this elementary process. 

The explanation why such a complicated reaction as the slow oxidation of phosphorus appears to be unimolecular is one which has often been brought forward in similar cases. The slowest part of the process is a diffusion, either of oxygen molecules to the reaction surface, or of phosphorus molecules from the surface, or of the inhibiting molecules of phosphorus oxide from the surface. According to the theory of diffusion the rates will follow the unimolecular law. Since the vapour pressure of phosphorus is low at ordinary temperatures, the... 

The result (14.114) gives the bimolecular rate k in terms of the intermolecular diffusion constant D and the intermolecular potential V r. The rate coefficient k associated with the reaction between the A and B species after they are assembled at the critical separation R is a parameter of this theory. If we regard the assembled A-B complex as a single molecule we could in principle calculate k as a unimolecular rate involving this complex, using the methodologies discussed in Sections 14.4 and 14.5. 

Most of the theory of diffusion and chemical reaction in gas-solid catalytic systems has been developed for these simple, unimolecular and irreversible reactions (SUIR). Of course this is understandable due to the obvious simplicity associated with this simple network both conceptually and practically. However, most industrial reactions are more complex than this SUIR, and this complexity varies considerably from single irreversible but bimolecular reactions to multiple reversible multimolecular reactions. For single reactions which are bimolecular but still irreversible, one of the added complexities associated with this case is the non-monotonic kinetics which lead to bifurcation (multiplicity) behaviour even under isothermal conditions. When the diffusivities of the different components are close to each other that added complexity may be the only one. However, when the diffusiv-ities of the different components are appreciably different, then extra complexities may arise. For reversible reactions added phenomena are introduced one of them is discussed in connection with the ammonia synthesis reaction in chapter 6. 

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