Reaction rate constant and regression

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. 

A reading of Section 2.2 shows that all of the methods for determining reaction order can lead also to estimates of the rate constant, and very commonly the order and rate constant are determined concurrently. However, the integrated rate equations are the most widely used means for rate constant determination. These equations can be solved analytically, graphically, or by least-squares regression analysis. 

Section 5.1 shows how nonlinear regression analysis is used to model the temperature dependence of reaction rate constants. The functional form of the reaction rate was assumed e.g., St = kab for an irreversible, second-order reaction. The rate constant k was measured at several temperatures and was fit to an Arrhenius form, k = ko exp —Tact/T). This section expands the use of nonlinear regression to fit the compositional and temperature dependence of reaction rates. The general reaction is... 

Although we cannot clearly determine the reaction order from Figure 3.9, we can gain some insight from a residual plot, which depicts the difference between the predicted and experimental values of cA using the rate constants calculated from the regression analysis. Figure 3.10 shows a random distribution of residuals for a second-order reaction, but a nonrandom distribution of residuals for a first-order reaction (consistent overprediction of concentration for the first five datapoints). Consequently, based upon this analysis, it is apparent that the reaction is second-order rather than first-order, and the reaction rate constant is 0.050. Furthermore, the sum of squared residuals is much smaller for second-order kinetics than for first-order kinetics (1.28 X 10-4 versus 5.39 xl0 4). 

Based on the computed AHf, ELUMO values, and the kinetic rates, linear tree-energy relationships (LFERs) for the dechlorination rate constants were developed by using a partial least squares (PLS) regression. Using this model, the reaction rate constants can be expressed as ... 

The computer program PROG1 determines the constants A and B from the regression analysis. Table 3-7 gives the results of the program with the slope -B equal to the reaction rate constant kx. Figure 3-19 shows a plot of In (D - D) against time t. 

Table 3 summarizes the results of the parameter estimation. All values of the kinetic parameters - reaction orders, rate constants and activation energies - were estimated by nonlinear regression analysis based on numerous experiments. 

The alkaline hydrolysis of phthalate diesters has been fit to the Taft-Pavelich equation (Eq. 9). Dimethyl phthalate (DMP) hydrolyzes to phthalic acid (PA) in two steps DMP + H20->MMP + CH30H and MMP + H20- PA + CH30H. The first step is about 12 times faster than the second, and nearly all the diester is converted to the monoester before product PA is formed. Other diesters are assumed to behave similarly. An LFER was obtained from rate measurements on five phthalate esters (Wolfe et al., 1980b). The reaction constants, p and S, were determined by multiple regression analysis of the measured rate constants and reported values of cr and Es for the alkyl substituents. The fitted intercept compares favorably with the measured rate constant (log kOH = — 1.16 0.02) for the dimethyl ester (for which a and s = 0 by definition). Calculated half-lives under pseudo-first-order conditions (pH 8.0, 30°C) range from about 4 months for DMP to over 100 years for di-2-ethylhexyl phthalate. 

Batch reactors are used primarily to determine rate law parameters for hotr geneous reactions. This determination is usually achieved by measuring cc centraiion as a function of time and then using either the differential, integr or nonlinear regression method of data analysis to determine the reacti order, a, and specific reaction rate constant, k. If some reaction parame other than concentration is monitored, such as pressure, the mole balance mi be rewritten in terms of the measured variable (e.g.. pressure as shown in t example in Solved Problems on the CD). 

Kinetic analysis Statistical data analysis was performed using the Statistica program version 6.0 (30). The usual kinetic models reported in literature to describe kinetic of compoimd formation are zero order [c= cO + kt], first order [c=cO exp (kt)] or second order [1/c = 1/cO + kt] reaction models. The Arrhenius equation k = kref exp (- Eai/R ( 1/T - 1 / Tref))] is usually applied to evaluate the effect of temperature on the reaction rate constant (31). For both levels of oxygen concentration a one step nonlinear regression method was performed and a regression analysis of the residuals was also carried out (32). 

Kj and Ej are the rate constant and the activation energy, respectively. A is a constant, R is the universal gas constant, and T is the absolute temperature. Thus, the kinetic model as discussed above allows calculation of activation energy (E), using linear regression on data obtained at different temperatures. Typical plots showing the effect of temperature on reaction rate and conversion of free radically polymerised unsaturated polyester are shown in Figures 1.2a and 1.2b, respectively. 

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)...

It is important to remember that in this day and age of powerful computers, it is no longer necessary to find analytical solutions to differential equations. Many commercially available software packages will cany out numerical integration of differential equations followed by nonlinear regression to fit the model, in the form of differential equations, to the data. Estimates of the rate constants and their variability, as well as measures of the goodness of fit of the model to the data, can be obtained in this fashion. Eventually, all modeling exercises are carried out in this fashion since it is difficult, and sometimes impossible, to obtain analytical solutions for complex reaction schemes. 

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