Rate state models phenomenology

If the above assumption is reasonable, then the modeling of most probable trajectories and of ensembles of trajectories is possible. We further discussed the calculations of the state conditional probability and the connection of the conditional probability to rate constants and phenomenological models. 

Phenomenologically, the rate and temperature dependence of the yield stress of semi-crystalline poljuners can be described by the Eyring activated state model, as discussed earlier, with either one (100,101) or two (46,102) activated processes. However, developing a theory for the yield of semi-crystalline polymers is clearly complicated by the presence of two distinct phases. It is imclear at present whether... 

The above phenomenological equations are assumed to hold in our system as well (after appropriate averaging). Below we derive formulas for P[Aq B, t), which start from a microscopic model and therefore makes it possible to compare the same quantity with the above phenomenological equa tioii. We also note that the formulas below are, in principle, exact. Therefore tests of the existence of a rate constant and the validity of the above model can be made. We rewrite the state conditional probability with the help of a step function - Hb(X). Hb X) is zero when X is in A and is one when X is ill B. 

Previous theoretical kinetic treatments of the formation of secondary, tertiary and higher order ions in the ionization chamber of a conventional mass spectrometer operating at high pressure, have used either a steady state treatment (2, 24) or an ion-beam approach (43). These theories are essentially phenomenological, and they make no clear assumptions about the nature of the reactive collision. The model outlined below is a microscopic one, making definite assumptions about the kinematics of the reactive collision. If the rate constants of the reactions are fixed, the nature of these assumptions definitely affects the amount of reaction occurring. 

In sharp contrast to the large number of experimental and computer simulation studies reported in literature, there have been relatively few analytical or model dependent studies on the dynamics of protein hydration layer. A simple phenomenological model, proposed earlier by Nandi and Bagchi [4] explains the observed slow relaxation in the hydration layer in terms of a dynamic equilibrium between the bound and the free states of water molecules within the layer. The slow time scale is the inverse of the rate of bound to free transition. In this model, the transition between the free and bound states occurs by rotation. Recently Mukherjee and Bagchi [14] have numerically solved the space dependent reaction-diffusion model to obtain the probability distribution and the time dependent mean-square displacement (MSD). The model predicts a transition from sub-diffusive to super-diffusive translational behaviour, before it attains a diffusive nature in the long time. However, a microscopic theory of hydration layer dynamics is yet to be fully developed. 

This monograph deals with kinetics, not with dynamics. Dynamics, the local (coupled) motion of lattice constituents (or structure elements) due to their thermal energy is the prerequisite of solid state kinetics. Dynamics can explain the nature and magnitude of rate constants and transport coefficients from a fundamental point of view. Kinetics, on the other hand, deal with the course of processes, expressed in terms of concentration and structure, in space and time. The formal treatment of kinetics is basically phenomenological, but it often needs detailed atomistic modeling in order to construct an appropriate formal frame (e.g., the partial differential equations in space and time). 

In this section, constitutive equations describing the polymerization kinetics when the system is in the liquid or rubbery state are analyzed. The influence of vitrification on reaction rate is considered in a subsequent section. First, phenomenological kinetic equations are analyzed then, the use of a set of kinetic equations based on a reaction model is discussed in separate subsections for stepwise and chainwise polymerizations. 

The above example gives us an idea of the difficulties in stating a rigorous kinetic model for the free-radical polymerization of formulations containing polyfunctional monomers. An example of efforts to introduce a mechanistic analysis for this kind of reaction, is the case of (meth)acrylate polymerizations, where Bowman and Peppas (1991) coupled free-volume derived expressions for diffusion-controlled kp and kt values to expressions describing the time-dependent evolution of the free volume. Further work expanded this initial analysis to take into account different possible elemental steps of the kinetic scheme (Anseth and Bowman, 1992/93 Kurdikar and Peppas, 1994 Scott and Peppas, 1999). The analysis of these mechanistic models is beyond our scope. Instead, one example of models that capture the main concepts of a rigorous description, but include phenomenological equations to account for the variation of specific rate constants with conversion, will be discussed. 

The rate constant (/ ,) was expressed in terms of the results of the computer simulations, for which a non-adiabatic transition-state theory (TST) model was used. Since the experimental results were analyzed in terms of a phenomenological Arrhenius model [158], we relate experiment (left-hand side) and theory (right-hand side) in terms of the following two equations. For the weakly temperature-dependent prefactor we have ... 

The reason for the nonexponential kinetics of solid-state chemical reactions lies in the existence of the rate constant distribution determined by the set of different configurations of the reactants, incommensurate to the solid lattice (see, e.g., ref. 177). In the case of low-temperature reactions the existence of the set of configurations can be phenomenologically accounted for by introduction of the equilibrium distance distribution R. As shown in the literature [138, 178, 179], introduction of this distribution into the discussed model of low-temperature reactions enables us to quantitatively describe the... 

As discussed in Section 8.1, a phenomenological stochastic evolution equation can be constructed by using a model to describe the relevant states of the system and the transition rates between them. For example, in the one-dimensional random walk problem discussed in Section 7.3 we have described the position of the walker by equally spaced points nAx (n = —cx3,..., oo) on the real axis. Denoting by Pin, Z) the probability that the particle is at position n at time Z and by kr and ki the probabilities per unit time (i.e. the rates) that the particle moves from a given... 

From a phenomenological point of view, it is natural to Interpret the multiexponential curves in Equations I, II, and III on the basis of energy cascading models. Such schemes assume - parallel to a single monomer and excimer state - additional electronic dwell-stations to be involved in serial energy relaxation processes. In a quantitative treatment, one has to diagonalize, then, rate equations of the form... 

It is important to clarify here that the description of PT processes by curve crossing formulations is not a new approach nor does it provide new dynamical insight. That is, the view of PT in solutions and proteins as a curve crossing process has been formulated in early realistic simulation studies [1, 2, 42] with and without quantum corrections and the phenomenological formulation of such models has already been introduced even earlier by Kuznetsov and others [47]. Furthermore, the fact that the fluctuations of the environment in enzymes and solution modulate the activation barriers of PT reactions has been demonstrated in realistic microscopic simulations of Warshel and coworkers [1, 2]. However, as clarified in these works, the time dependence of these fluctuations does not provide a useful way to determine the rate constant. That is, the electrostatic fluctuations of the environment are determined by the corresponding Boltzmann probability and do not represent a dynamical effect. In other words, the rate constant is determined by the inverse of the time it takes the system to produce a reactive trajectory, multiplied by the time it takes such trajectories to move to the TS. The time needed for generation of a reactive trajectory is determined by the corresponding Boltzmann probability, and the actual time it takes the reactive trajectory to reach the transition state (of the order of picoseconds), is more or less constant in different systems. 

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