Reaction rate constant and regression coefficient

Table 23.1 HMF formation kinetics in isothermal heating as a function of treatment temperature, first order reaction pseudo rate constant and regression coefficients...

A further relationship between reaction rate and sty has been preposed by Labuza. A linear relationship between the logarithms of reaction rate constants and ay was found to exist for many foods within a range of ay 0.3 to 0.8. Figure 4 depicts this relationship for the rate of formation of pyrazine and 2-methylpyrazine as a function of sty. Table V lists the regressions determined for the three rate constants which fell within this range. Correlation coefficients (r2) for these regressions were quite high, at 0.99 for both pyrazine and 2-methylpyrazine. 

Equations (2) and (3) were fit to experimental data using nonlinear regression to obtain values of the first-order reaction rate constants and the stoichiometric coefficients at each temperature. The conversion data from the 400°C thermal run and the best fit of the kinetic model are shown in Figure 1. It is interesting to note that at the time of incipient coke formation ( 60 minutes) the asphaltene and maltene data deviate from predicted first-order behavior. From this we concluded that both asphaltenes and maltenes were participating in secondary coke-forming reactions. Further separation of the maltenes into resins (polar aromatics) and oils confirmed this to be true and showed that it was the resin fraction that was involved in coke formation. 

Note the use of activities, as well as of an equilibrium constant based on activities. The kinetic constants for autocatalyzed and catalyzed reactions, k and k, were determined from initial reaction rates with liquid activity coefficients calculated by UNIQUAC. Near chemical equilibrium the fCT is about 6, while Kx is about 5. Table 8.7 gives activation energies and pre-exponential factors obtained by nonlinear regression. The simulation shows tbat the autocatalysis effect is neghgible below 150 °C, but it might increase to 20% at 180 °C. 

Reaction rate constants, k, and adsorption coefficients, Ki, obtained from linear and nonlinear regression analyses must be positive. 

Table 1. Reaction rate constant k (M s ) and regression coefficient r of fumigants and ammonium thiosulfate in aqueous phase at 20°C (28)...

The present system showed a linear plot of DP as a function of reaction time. The relationship of such stepwise order reactions is given by Equation 2, where [PtCl42] = [MTX] = [Ao ] and [Ao ] is the starting molar concentration, k is the observed rate constant, and t is the reaction time. Regression analysis shows a linear relationship with a correlation coefficient of 0.998 and an observed rate constant of k = 1.0 X 10 2 liter mole sec- (Figure 1). The Y-intercept for the system is not one, as suggested by Equation 3. This indicates additional factors are involved. 

Once the reactor equations and assumptions have been established, and HDS, HDN, HDA, and HGO reaction rate expressions have been defined, the adsorption coefficient, equilibrium constants, reaction orders, frequency factors, and activation energies can be determined from the experimental data obtained at steady-state conditions by optimization with the Levenberg-Marquardt nonlinear regression algorithm. Using the values of parameters obtained from steady-state experiments, the dynamic TBR model was employed to redetermine the kinetic parameters that were considered as definitive values. The temperature dependencies of all the intrinsic reaction rate constants have been described by the Arrhenius law, which are shown in Table 7.4. 

The rate of H2Oz consumption and the OH production were directly related to total iron concentration. The concentrations of hydroxyl radical produced were controlled by the rate of reaction with dissolved constituents. Rate constants for adsorption (ka) and desorption (kd) of PCBs from particles were calculated by regression of data from 1.5 to 5 hr. Adsorption rate constants were estimated from Equation (6.130) assuming that the partitioning rate constants between 2 and 5 hr without OH could be used for calculation of equilibrium partition coefficients Ky)... 

Oxygenation of 2-substituted adamantanes with methyl(trifluoromethyl)dioxirane showed a reaction constant, p = -2.31, consistent with a strongly electron-demanding transition state. Analysis of the effect of solvents on the rate yielded a positive regression coefficient with Dimroth-Reichardt Ej solvent polarity parameter. A mechanism involving an electrophilic O atom insertion has been postulated for the formation of alcohols and carbonyl compounds.201... 

Both the and the coefficients in the linear combination relations are functions of the rate constants, through the matrix transformations. Obviously, Eq. 1.6.2-1 is enormously isier to use in determination of the than the full solutions for the y which consist of Af-exponential terms, and which would require nonlinear regression techniques. In foct, simple logarithmic plots, as just described, can be used. Once the straight-line reaction paths are used to determine the numerical matrix manipulations can then be used to readily recover the kj,. ... 

For a simple order, the rate expression can be integrated and special plots utilized to determine the rate coefficient. A plot of MCa versus t or xjj -x versus t is used similarly for a second-order irreversible reaction. For reversible reactions with first order in both directions a plot of ln(Ci - Qeq)/(Qo - Qeq) or ln(l - XaIxa versus f yields ( i + ki) from the slope of the straight line. Using the thermodynamic equilibrium constant K = kxlkz, both k and kz are obtained. Certain more complicated reaction rate forms can be rearranged into such linear forms. These plots are useful for an estimate of the "quality" of the fit to the experimental data and can also provide initial estimates to formal regression techniques that will be systematically discussed in Chapter 2. 

The rate constants ki were subjected to correlation analyses using various single- and multi-parametric equations the best correlation was obtained using Charton s LDR and LDRS equations. The polar regression coefficients are negative, indicating the formation of a cationic species in the transition state. The reaction is subject to steric hindrance by orf/to-substituents. ... 

The constant K, should thus not be confused with the surface exchange coefficient defined by Equation (14.21). Both parameters have the same dimension (cm s ) A general method of regression analysis of data from relaxation experiments using the linear rate law for the surface reaction has been given. Other empirical rate laws for the surface exchange reaction have been proposed by Dovi et al. and Gesmundo et al. °... 

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