A phenomenological approach

An irreversible reaction between the reactive solutes A and B in an inert solvent can be pictured to proceed in two stages the approach of A and B from a large separation distance to a distance at which chemical forces become dominant and the subsequent reaction of A and B when in this close configuration. Schematically, it can be described as 

The three rate coefficients indicated in eqn. (1) are those for the formation of the psuedo species (AB) from separated A and B, fed, the backward rate coefficient for this step, k d, and the first-order rate coefficient, fea, for the reaction of (AB) to form products. Depending on the nature of the interaction between A and B once they come close to each other, (AB) might be described by several terms. If the psuedo species (AB) is at the saddle point of the potential energy surface between A + B, it would be described as an activated complex. When A and B are in contact, but little or no interaction occurs between them, then the term contact or collision pair is appropriate. Usually, the exact details of the interaction energy between A and B are not known, but will probably be between 

Reverting to eqn. (1), the overall rate coefficient for the formation of products, providing that the concentration of the encounter pair (AB) is small and assuming a steady state, is kAka/(k-d + ka). When the reaction of the encounter pair to form products becomes fast compared with the diffusive forward and backward processes, the rate of formation of products is determined by the rate at which A and B can diffuse together to form the encounter pair (AB). The opposite situation is where the rate of reaction of the encounter pair is very slow compared with the backward diffusion (ka fe d), and the products form at a rate 

The recombination of methyl or ethyl radicals in the gas phase or in iso-octane solution have very similar rate coefficients [22]. In the gas phase, the number of collisions between, for instance, methyl radicals of concentration n and molecular diameter a, is cj2 where c is the 

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For modelling conformational transitions and nonlinear dynamics of NA a phenomenological approach is often used. This allows one not just to describe a phenomenon but also to understand the relationships between the basic physical properties of the system. There is a general algorithm for modelling in the frame of the phenomenological approach determine the dominant motions of the system in the time interval of the process treated and theti write... 

Values of kH olki3. o tend to fall in the range 0.5 to 6. The direction of the effect, whether normal or inverse, can often be accounted for by combining a model of the transition state with vibrational frequencies, although quantitative calculation is not reliable. Because of the difficulty in applying rigorous theory to the solvent isotope effect, a phenomenological approach has been developed. We define <[), to be the ratio of D to H in site 1 of a reactant relative to the ratio of D to H in a solvent site. That is. 

There is a lively controversy concerning the interpretation of these and other properties, and cogent arguments have been advanced both for the presence of hydride ions H" and for the presence of protons H+ in the d-block and f-block hydride phases.These difficulties emphasize again the problems attending any classification based on presumed bond type, and a phenomenological approach which describes the observed properties is a sounder initial basis for discussion. Thus the predominantly ionic nature of a phase cannot safely be inferred either from crystal structure or from calculated lattice energies since many metallic alloys adopt the NaCl-type or CsCl-type structures (e.g. LaBi, )S-brass) and enthalpy calculations are notoriously insensitive to bond type. 

Two different approaches have been followed to calculate the lineshapes within a relaxation model. According to a phenomenological approach based on the modified Bloch equations [154, 155], the intensity distribution of the theoretical Mossbauer spectrum may be written as [156] ... 

TG Kaufmann, EF Leonard. Studies of intramembrane transport A phenomenological approach. AIChE J 14 110-117, 1968. 

The hydrodynamic interaction is introduced in the Zimm model as a pure intrachain effect. The molecular treatment of its screening owing to presence of other chains requires the solution of a complicated many-body problem [11, 160-164], In some cases, this problem can be overcome by a phenomenological approach [40,117], based on the Zimm model and on the additional assumption that the average hydrodynamic interaction in semi-dilute solutions is still of the same form as in the dilute case. 

W. B. Davis, M. R. Wasidewski, M. A. Ratner, V. Mujica, A. Nitzan, Electron Transfer Rates in Bridged Molecular Systems - a Phenomenological Approach to Relaxation , J. Phys. Chem. 1997,101, 6158-6164. 

Pettifor s structure maps additional remarks. We have seen that in a phenomenological approach to the systematics of the crystal structures (and of other phase properties) several types of coordinates, derived from physical atomic properties, have been used for the preparation of (two-, three-dimensional) stability maps. Differences, sums, ratios of properties such as electronegativities, atomic radii and valence-electron numbers have been used. These variables, however, as stressed, for instance, by Villars et al. (1989) do not always clearly differentiate between chemically different atoms. 

Some other theoretical aspects of ionic solvation have been reviewed in the last few years. The interested reader is referred to them ionic radii and enthalpies of hydration 20>, a phenomenological approach to cation-solvent interactions mainly based on thermodynamic data 21>, relationship between hydration energies and electrode potentials 22>, dynamic structure of solvation shells 23>. Brief reviews, monographs, and surveys on this subject from a more or less different point of view have also been published 24—28) ... 

In general there is no obvious way of separating the adsorbent molecule into two well-defined subunits, nor can one assume that all interactions between the ligand and the adsorbent molecule consist of only two electrostatic interactions. This is clearly the case for succinic acid or for DNA, where there is an infinite number of ways to define an unperturbed intrinsic binding constant. Therefore, the use of such factorization of should be abolished when a phenomenological approach is adopted. On the other hand, when using a molecular approach one may or may not use such factorization. In any case this is certainly not a necessity. 

Nowadays attention is turned also to the supermolecular level, that is, to the morphologic aspects, to the nature of interfaces, to the formation of new phases, or of particular aggregates (liquid crystals, gels, etc.). Interest has also been directed to the study of chain mobility for its influence on frictional properties of polymers. In recent years there have been many successful approaches to a microscopic theory (in contrast to a phenomenological approach) of the physi-comechanical behavior of macromolecular materials. 

The evaluation of the I K) term in Eq. (38) beeomes inereasingly difficult for more complex target and projectile systems. This encouraged the development of a phenomenological approach for use with such systems. In the method proposed by Toburen et al. [71], one considers the screening of the projectile by its bound electrons as a function of the distance R from the nucleus as... 

A typical dipole moves in a potential well without friction during the lifetime tv between two consecutive strong collisions. The well, the form of which should preferably be taken as simple as possible, is chosen on the basis of a phenomenological approach based on the experience gained in previous studies (VIG, GT). The time tv is interpreted as a duration of local order in a liquid. The exponential distribution over fv is used, where the mean lifetime r (which later is referred to simply as lifetime ), r = (fv), is a free parameter of the model. 

In Section VII we have applied a phenomenological approach in point of the forms of the potential wells characteristic for water. Now we shall consider a principally another way of modelling of intermolecular interactions pertinent to vibration of the H-bonded molecules. Recently [10, 12, 12a], a preliminary studies of the molecular dynamics was undertaken based on some picture (although very crude) of the molecular structure of water. We shall here briefly represent these results, namely ... 

We have followed a phenomenological approach and used the cluster model [18]. In this model the eg-type distortion interacts more strongly with the electronic state of an octahedral coordinated Cr3+ ion than the distortions of t2g symmetry. According to Ham [19], we assume that the continuum of vibrational modes with eg character can be approximated by a single mode with an effective frequency o>, mass /r and coupling constant V. The collective coordinates of the eg mode are conventionally known as Qe x2 — y2) and Qs ( 3z2 — r2). The linear Jahn-Teller Hamiltonian in equation (1) for the X state is [18] ... 

Following a phenomenological approach, the driving force for chemical diffusion, in oxides, is the gradient of the electrochemical potential, ri-... 

Polarization model for the hydration force a phenomenological approach.32... 

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