Kramers theory applications

E. Poliak The RRKM theory and Kramers theory and its later generalizations by Grote, Hynes, and other are two sides of the same coin. In the spatial diffusion limit, one can show that Kramers s rate expression is identical in form to the RRKM expression, that is, a ratio of equilibrium unidirectional flux and density of reactants. The difficult problem in the application of RRKM theory to the stilbene molecule with a few attached benzenes is whether the equilibration of energy occurs fully on the time scale of the isomerization. One should also... 

The critical nucleus of a new phase (Gibbs) is an activated complex (a transitory state) of a system. The motion of the system across the transitory state is the result of fluctuations and has the character of Brownian motion, in accordance with Kramers theory, and in contrast to the inertial motion in Eyring s theory of chemical reactions. The relationship between the rate (probability) of the direct and reverse processes—the growth and the decrease of the nucleus—is determined from the condition of steadiness of the equilibrium distribution, which leads to an equation of the Fourier-Fick type (heat conduction or diffusion) in a rod of variable cross-section or in a stream of variable velocity. The magnitude of the diffusion coefficient is established by comparison with the macroscopic kinetics of the change of nuclei, which does not consider fluctuations (cf. Einstein s application of Stokes law to diffusion). The steady rate of nucleus formation is calculated (the number of nuclei per cubic centimeter per second for a given supersaturation). For condensation of a vapor, the results do not differ from those of Volmer. 

Recently much attention has been aroused on solution reactions whose rates decrease as the viscosity Tj of solvents increases. These reactions cannot be rationalized in the framework of the transition state theory. To describe them, two currents of theories have been developed by extending the Kramers theory. One was initiated by Grote and Hynes, while the other by Sumi and Marcus. Recent data on thermal Z/E isomerization of substituted azobenzenes and A/ -benzyU-deneanilines confirms the applicability of the latter for 77 variation over 10 times under pressure. 

This chapter reviews the generalizations of the Kramers model that were develojjed during the past few years. The result of this effort, which we may call the generalized Kramers theory, provides a useful framework for the theoretical description of activated rate processes in general and of chemical reaction rates in condensed phases in particular. Some applications of this framework as well as its limitations are also discussed. In the last few years there has also been substantial progress in the study of the quantum mechanical Kramers model, which may prove useful for condensed phase tunneling reactions. This aspect of the problem is not covered by the present review. 

The Kramers theory and its extensions have found many applications since the original work by Kramers. Recent application of the non-Markovian theory in the low-friction limit to thermal desorption was described by Nitzan and Carmeli. Another novel application of the Markovian theory is to transition from a nonequilibrium state of a Josephson junction. In what follows we shall briefly review the recent application of the generalized Kramers theory to chemical rate processes. More detailed reviews of the exjjerimental and theoretical status of this field may be found in Hynes. ... 

SCEATS - In your work you have considered the Kramers model for 1-dimensional motion in the high and low friction limits and the Smoluchowski model for 3-dimensional motion. In my paper at this conference, I will show that the 3-d motion can be reduced to a one-dimensional problem by use of the one dimensional potential V(R)-2kTlnR, and that application of Kramers theory to the dynamics on this potential yields the Smoluchowski-Debye result in the high friction limit. Hence unimolecular and bimolecular reactions can be compared using the same model. 

J. R. Kramer and H.E. Allen (eds), Metal Speciation Theory, Analysis and Application, Lewis, Chelsea (1988). 

Kramer, S. (2001). Liposome/water partitioning theory, techniques, and applications. In Pharmacokinetic Optimization in Drug Research Biological, Physicochemical, and Computational Strategies, eds. Testa, B., van Waterbeemd, H., Folkers, G. and Guy, R., Series, Yerlag Helvetica Chimica Acta, Zurich. 

Per-Olov Lowdin had a long and lasting interest in the analytical methods of quantum mechanics and my tribute to his legacy involves an application of the Wentzel-Kramers-Brillouin (WKB) asymptotic approximation method. It was the subject of a contribution(l) by Lowdin to the Solid State and Molecular Theory Group created by John C. Slater at the Massachusetts Institute of Technology. 

Early applications of WKB approximations to the Coulomb problem in Schrodinger theory demonstrated the necessity and expediency of the Kramers modification ) ... 

Oilman, L.-O. and Sjoberg, S. (1988) Thermodynamic calculations with special reference to the aqueous aluminium system. In Metal Speciation Theory, Analysis and Application (eds Kramer, J.R. and Allen, H.E.). Lewis Publishers, Chelsea, MI, pp. 1-40. 

The theory of Brownian motion is a particular example of an application of the general theory of random or stochastic processes [2]. Since Kramers approach is based on a more general stochastic equation than the Langevin equation, we have reviewed some of the fundamental ideas and methods of the theory of stochastic processes in Appendix H. 

Comment In the case of solvent dynamics in chemical reactions, the theory of Kramers was already available, but there were no experiments for its application at that time and perhaps as a result the theory remained largely unused by chemists for many decades. Later, however, experiments did... 

Kramer SD. Liposome/water partitioning theory, techniques and applications. In Pharmacokinetic Optimization in Drug Research. 

Ohman, L. O., and Sjoberg, S. (1988) Thermodynamic Calculations with Special Reference to the Aqueous Aluminum System. In Metal Speciation Theory, Analysis and Application, J. R. Kramer and H. E. Allen, Eds., Lewis, Chelsea, MI. O Melia, C. R. (1972) Coagulation and Flocculation. In Processes for Water Quality Control, W. J. Weber, Jr., Ed., Wiley-Interscience, New York, pp. 61-110. O Melia, C. R. (1987) Particle-Particle Interactions. In Aquatic Surface Chemistry, W. 

Although the Kramers model contains much of the essential physics of the activated escajje problem, it cannot be used for quantitative discussion of many realistic activated processes. In particular the model is too oversimplified for the original application intended by Kramers for chemical rate processes. The theory needs to be generalized to correct the following shortcomings of the Kramers model. 

Tessier. A. and Campbell. P.G.C.. Partitioning of trace metals in sediments In metal speciation. In Theory, analysis and application, J.R. Kramer and H. E. Allen (Eds). Lewis Publisher. 

Besides the diseontinuous states there are also wstates forming a continuous range (with positive energy) they correspond to the hyperbolic orbits of Bohr s theory. The jumps from one hyperbola to another or to a stationary state give rise to the emission of the continuous X-ray spectrum emitted when electrons are scattered or caught by nuclei. The intensity of this spectrum has been calculated by Kramers (1923) from the standpoint of Bohr s theory by a very ingenious application of the correspondence principle. His... 

Presently, it will be a concern to review the basics of crystal field theory as a vehicle to understand the electronic features of transition metal atoms and ions in an octahedral environment. Thus is considered the limited basis of ten spinorbitals of the partially occupied atomic d-shell for the relevant transition metal. A particular choice of basis is made in order to obtain a convenient form for the spin-orbital interaction and to simplify the application of the point group symmetry. The e-type orbitals spin factors, a basis for the four-dimensional irreducible representation U. The Kramers pairs will be used ... 

Our approach to the study of the departure from equilibrium in chemical reactions and of the "microscopic theory of chemical kinetics is a discrete quantum-mechanical analog of the Kramers-Brownian-motion model. It is most specifically applicable to a study of the energy-level distribution function and of the rate of activation in unimolecular (dissociation Reactions. Our model is an extension of one which we used in a discussion of the relaxation of vibrational nonequilibrium distributions.14 18 20... 

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