Population kinetics distribution

A kinetic study for the polymerization of styrene, initiated with n BuLi, was designed to explore the Trommsdorff effect on rate constants of initiation and propagation and polystyryl anion association. Initiator association, initiation rate and propagation rates are essentially independent of solution viscosity, Polystyryl anion association is dependent on media viscosity. Temperature dependency correlates as an Arrhenius relationship. Observations were restricted to viscosities less than 200 centipoise. Population density distribution analysis indicates that rate constants are also independent of degree of polymerization, which is consistent with Flory s principle of equal reactivity. 

The resin system selected to initiate these studies is a step-growth anhydride cured epoxy. The approach to the kinetic analysis is that which is prevalent in the chemical engineering literature on reactor design and analysis. Numerical simulations of oligomeric population density distributions approximate experimental data during the early stages of the cure. Future research will... 

An experimental setup identical to that for PHOFEX spectra can be used to measure correlations between two scalar (internal state population, kinetic energy) quantities or between a scalar and a vector quantity (angular distribution of fragment rotation J or velocity v). By holding the PROBE laser frequency fixed and scanning the MONITOR laser through the Doppler pro-... 

In any event, let us examine low-temperature fluorescence results from a magnesium-porphyrin myoglobin (Fig. If the reader actually consults the cited works, he/she may be confused by the term population distribution function (PDF) used by the authors, which is easily confused with the phonon wing (PW) discussed up to this point. Vanderkooi et al. use the term PDF as an acronym for the conformational distribution that Frauenfelder et al. have used to explain low-temperature reaction kinetic distributions, and they measure this distribution function by picking out the variation in a ZPL intensity with excitation wavelength. 

D. Population Kinetic Analysis Uses for Prior Parameter Distributions... 

Unfortunately, there has been minimal work on identifiability issues with respect to population kinetic analysis. The current work on identifiability of kinetic models (Jacquez, 1985 belli and DiStefano, 1980 Cobelli and Saccomani, 1990) focuses on estimation for an individual experiment. With regard to population analysis and Bayesian estimation of parameters for an individual from the population, it is the population analysis step for which identifiability issues need to be considered. Once a prior distribution... 

In elementary gas kinetic theory, a gas is modeled as a collection of many randomly moving noninteracting molecules. Each molecule is assumed to obey classical mechanics with its state specified by its position and velocity. We treat the molecules in a gas as a population. The distribution for the x component of the velocity of a molecule is the following Gaussian distribution ... 

Correlations of nucleation rates with crystallizer variables have been developed for a variety of systems. Although the correlations are empirical, a mechanistic hypothesis regarding nucleation can be helpful in selecting operating variables for inclusion in the model. Two examples are (/) the effect of slurry circulation rate on nucleation has been used to develop a correlation for nucleation rate based on the tip speed of the impeller (16) and (2) the scaleup of nucleation kinetics for sodium chloride crystalliza tion provided an analysis of the role of mixing and mixer characteristics in contact nucleation (17). Pubhshed kinetic correlations have been reviewed through about 1979 (18). In a later section on population balances, simple power-law expressions are used to correlate nucleation rate data and describe the effect of nucleation on crystal size distribution. 

Population balances and crystallization kinetics may be used to relate process variables to the crystal size distribution produced by the crystallizer. Such balances are coupled to the more familiar balances on mass and energy. It is assumed that the population distribution is a continuous function and that crystal size, surface area, and volume can be described by a characteristic dimension T. Area and volume shape factors are assumed to be constant, which is to say that the morphology of the crystal does not change with size. 

The population balance analysis of the idealized MSMPR crystallizer is a particularly elegant method for analysing crystal size distributions at steady state in order to determine crystal growth and nucleation kinetics. Unfortunately, the latter cannot currently be predicted a priori and must be measured, as considered in Chapter 5. Anomalies can occur in the data and their subsequent analysis, however, if the assumptions of the MSMPR crystallizer are not strictly met. 

Several authors have presented methods for the simultaneous estimation of crystal growth and nucleation kinetics from batch crystallizations. In an early study, Bransom and Dunning (1949) derived a crystal population balance to analyse batch CSD for growth and nucleation kinetics. Misra and White (1971), Ness and White (1976) and McNeil etal. (1978) applied the population balance to obtain both nucleation and crystal growth rates from the measurement of crystal size distributions during a batch experiment. In a refinement, Tavare and... 

Nucleation in a pure liquid. According to the kinetic theory for pure gases and liquids, there are local fluctuations of densities, which are clusters of molecules in a gas and holes (or vapor clusters) in a liquid. Frenkel (1955) established the population distribution of such holes of phase B in a liquid of continuum phase A by Boltzmann s formula,... 

In some cases, two or more catalysts are present during polymerization. Inevitably, the catalysts exhibit different polymerization kinetics, which results in different populations of molecules. In such cases, we produce polymers with a bimodal molecular weight distribution. 

If thermal motion on the Ti (or Si) surface leads to a quasi-equilibrium distribution of molecules between several minima, some of them are likely to provide a faster return to So than others and they will then drain the excited state population and determine which products will be formed. This is a straight-forward kinetic problem and it is clear that the process need not be dominated by the position of the lowest-energy accessible minimum in the excited hypersurface. Such minima may correspond to conformers, valence isomers, etc. Of course, it is well known that ground-state conformers may correspond to excited-state isomers, which are not in fast equilibrium. 65,72) Also, there is no reason why several separate minima in Si or Ti could not correspond to one minimum in So, and there is some evidence that this situation indeed occurs in certain polycyclic cyclohexenones. 73,74)... 

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