Analysis and Correlation of Kinetic Data

After completing this chapter, you should be able to 

linearize rate equations, i.e., put them into a straight-line form  

test linearized rate equations against experimental data graphically, and obtain preliminary estimates of the unknown parameters in the rate equation  

obtain best fit values of the unknown parameters in a linearized rate equation using linear least-squares analysis  

obtain best fit values of die parameters in a nonlinear rate equation using nonlinear least squares analysis  

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The application of NMR chemical shifts, 5n, of nitrogen atoms in nitro groups to the analysis and prediction of kinetic data of thermal decomposition of nitramines [10] is presented in Fig. 1 a non-isochronous molecule HNIW correlates through the value in the position 2. 

Predicting the solvent or density dependence of rate constants by equation (A3.6.29) or equation (A3.6.31) requires the same ingredients as the calculation of TST rate constants plus an estimate of and a suitable model for the friction coefficient y and its density dependence. While in the framework of molecular dynamics simulations it may be worthwhile to numerically calculate friction coefficients from the average of the relevant time correlation fiinctions, for practical purposes in the analysis of kinetic data it is much more convenient and instructive to use experimentally detemiined macroscopic solvent parameters. 

The parameters Ci, t2 were postulated to be dependent only upon the substrate, and d, d2, upon the solvent. A large body of kinetic data, embodying many structural types and leaving groups, was subjected to a statistical analysis. In order to achieve a unique solution, these arbitrary conditions were imposed cj = 3.0 C2 for MeBr Cl = C2 = 1.0 for f-BuCl 3.0 Ci = C2 for PhsCF. Some remarkably successful correlations [calculated vs. experimental log (fc/fco)l were achieved, but the approach appeared to lack physical significance and was not much used. Many years later Peterson et al. - showed a correspondence between Eqs. (8-69) and (8-74) in particular, the very simple result di + d, = T was found. 

In these later sections, interpretations of quantitative data for product mixtures are emphasised, and the relationship between kinetics and product analysis will be developed. Mechanistic applications of kinetic data are limited to steps of reactions prior to and including the rate-determining step. As separate later steps often determine the reaction products, detailed product studies and investigations of reactive intermediates are important supplements to kinetic studies. Examples of solvolytic and related (SN) reactions have been chosen first because they provide a consistent theme, and second because SN reactions provide an opportunity to assess critically many of the mechanistic concepts of organic chemistry. Product composition in solvolytic reactions will be discussed next followed by product selectivities (Section 2.7.2) and rate-product correlations (Section 2.7.3). 

In 2001, Holzwarth et al. [125, 136] stressed the importance of transient studies as an alternative to steady continuous reactor operations. A combination of micro kinetic analysis together with transient experiments allowed the determination of the global catalytic conversion from elementary reaction steps. A prerequisite for such an analysis is the correlation of experimental data with the data of a model. In Figure 3.85, experimental and model responses of an impulse of reactants were correlated. Agreement between the data helped to derive the reaction mechanism. 

Interpretation of available data is frustrated by lack of knowledge of certain fundamental quantities such as Interfacial area, mass transfer coefficients, solubility data, diffusion coefficients, bubble sizes, etc.. Existing equations for almost all of these variables have been developed on the basis of experiments conducted at atmospheric pressure and around room temperature. Use of such predictive equations at the reacting conditions involves large extrapolation, and the combined errors would make the analysis of kinetic data very suspect. In spite of this, most work reported in the literature does use such correlations. 

Analysis of Kinetic Data. To gather the data necessary for this type of analysis, yet another set of phenolysis experiments was carried out this time in a small Parr bomb that was immersed in an oil bath maintained at 220 °C for relatively short periods of time. The data collected and summarized in Table VI were correlated for zero, first, and second order kinetic models by the use... 

Such an analysis, however, necessarily Involves the acquisition and correlation of data related to the viscosity and kinetics of the curing polymer. Computer modeling of mold filling and curing events is then possible (2.,. Coupling these... 

That equation and its integral form soon gained enormous popularity among scientists investigating the kinetics of gas adsorption on solid surfaces. Correlations of experimental data usually started with the application of that equation, and the frequently noted deviations from "Elovich behavior" were the form of analysis in dozens of published papers. Even reviews followed that general trend, discussing the Elovich equations and "deviations" from it. 

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. 

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