Arrhenius regression analysis

The large deviation between the estimated rate constants for a given temperature extrapolated from classical (Fig. 35) and modified Arrhenius regression analysis (Fig. 37) and the experimental rate constants determined at the actual temperature is due to the linear regression analysis method used. That is, the values of the logarithms of the rate constants do not reflect directly the errors in the experimental data. As Bentley pointed out,317 when the error term e is added to the logarithm of k in accordance with Eq. (2.76), in the linear... 

The activation energy differences of My as well as of and M, and k /kp and kt/kp. were calculated from Arrhenius and Mayo plots, respectively, by linear regression analysis using a computer. Hie AEjjw values given in kcal/mole can be converted to kJ/mole by multiplying with 4.18. 

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... 

Similar plots have been obtained for the gas-phase rearrangement of 35 (A = CH3 Aik = ethyl Alk = methyl) and 36 (A = CH3 Aik = methyl Alk = ethyl) in 720 torr methyl chloride in the temperature range from 40 to 120 Regression analysis of the relevant Arrhenius curves leads to the activation parameters listed in Table 22. 

Comparisons made below refer to kinetic data obtained for processes proceeding under similar conditions. All available values of (log A, E) within each group of related reactions were included in the linear regression analysis (Appendix II) and the compensation line was calculated using these formulas. Unless otherwise stated, the units of A are always molecules m-2 sec-1 at 1 Torr pressure of reactants and those of E are kJ mole-1. The compilation of Arrhenius parameters referred to identical reaction conditions is not always easy (or, indeed, possible in some instances) and it may be necessary to recalculate data from literature sources using an extrapolation. Not all details of the necessary corrections are recorded below, but such estimations were always minimized to preserve the objectivity of the conclusions reached. 

In 1991 Moffat, Jensen and Carr employed RRKM theory in the form of a nonlinear regression analysis of experimental data to estimate the high-pressure Arrhenius parameters for elimination of H2 from SM4 as logA = 15.79 0.5 s 1, E = 59.99 2.0 kcalmor1 and A//j(SiH2) = 65.5 1.0 kcalmorl62. 

Figure 7.23. Effects of temperature on mitochondrial function. (Upper panel). Arrhenius plot illustrating the slope discontinuity ( break ) that commonly occurs at a high temperature of measurement, the Arrhenius break temperature (ABT). Data are for mitochondria of the hydrothermal vent tubeworm Riftia pachyptila (after Dahlhoff et al., 1991). (Lower panel) Arrhenius break temperatures for mitochondrial respiration of diverse invertebrates and fishes. The open square is for mitochondrial respiration of the Antarctic nototheniid fish Trematomus bernacchii and is not included in the regression analysis. A line of identify (ABT = adaptation temperature) is also shown (see text for analysis). (Data from Dahlhoff and Somero, 1993b Dahlhoff et ah, 1991 Weinstein and Somero, 1998.)...

The Arrhenius equation did not describe very well the influence of temperature on viscosity data of concentrated apple and grape juices in the range 60-68 °Brix (Rao et al., 1984, 1986). From non-linear regression analysis, it was determined that the empirical Fulcher equation (see Ferry, 1980 p. 289, Soesanto and Williams, 1981) described the viscosity versus temperature data on those juice samples better than the Arrhenius model (Rao et al., 1986) ... 

Figure 38. Linear Arrhenius regression by weighted least-squares analysis (a) and least-squares analysis (b). y -axis (1) logarithmic scale, (2) arithmetical scale. (Reproduced from Ref. 317 with permission.)...

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). 

Our previous MNR Arrhenius parameters have been revised based on weighted linear regression analysis. The present temperature corrections have been based on the new results, which have been included in Appendix A. 

The modified Arrhenius parameters are determined from regression analysis with application of the principle of least squares. CTST describes the forward rate constant from reactant to the transition state (TS) s a function of the equilibrium between reactant and TS. ThermKin requires thermodynamic properties in the NASA polynomial format, needs to know whether the reaction is uni- or bimolecular, and either a two-parameter fit or a three-parameter fit is desired. Finally, the reaction to be calculated has to be given in the form ... 

Methodology 1 involved plotting the log of the rate constants determined at pH 3.75 at 190°C, 200 C and 210 C (tom Table I) versus 1/Temperature (Kelvin), lliis Arrhenius plot is shown in Figure 5. The calculated was 52.6 Kcal/mole (r = 0.931) and was found from the slope of the line generated by regression analysis tom the following relationship ... 

Creep - The creep model used in CARES combines the classical Norton isothermal power law with a thermal process of constant activation energy showing the Arrhenius type of temperature dependence. Material creep data is determined by a log linear regression analysis of specimen steady state creep data determined at a variety of temperatures and stress levels. Similar to the oxidation analysis, data is examined by CARES to determine if it contains outliers. Any outliers are deleted prior to the final determination of the three material constants. 

The time, at each test temperature, that reduces a physical property to 50% of its original value is then plotted, and an Arrhenius curve is fitted to the data points by regression analysis as illustrated in Fig. 3-28. 

The chemistry for a stoichiometrically balanced reaction suggests that m = 1 and n = 2 in Eq. (2.86). For real systems, values are often close to these values but not identical. In the epoxy-amine reaction the alcohol, which may be present initially in small concentrations but is also a product of the reaction, catalyzes further reaction, resulting in autocatalysis. Since there are four unknowns ki, k2, m, and n) nonlinear regression analysis must be employed, although ki can be evaluated independently as the extrapolated reaction rate at a = 0. Autocatalytic kinetics are usually evaluated by the derivative form of the autocatalytic rate equation [Eq. (2.86)] with data coUected by isothermal method 1 measurements. Activation energy E and preexponential factor A are measured from the Arrhenius equation... 

For the regression analysis, a reparameterized form of the Arrhenius and Van t Hoff equations was used. A full statistical analysis, which included the calculation of the 95% confidence intervals on the estimated parameters, was performed after regression. Initially, the response curves were recorded using a sampling time of 1 ms. For a correct statistical analysis the sampling time had to be increased to 8 ms. 

In 1966 we noted that a log tJq - plot published by Spencer and Dillon for an ar-PS fraction consisted of two straight lines intersecting near T . Later studies by ourselves using both first derivatives and regression analysis with residuals revealed double or triple Arrhenius plots (depending on the temperature range of the data) for ar-PSs of 1.1, for a broad... 

Figure 4.27. Arrhenius analysis The right-hand panel shows the assay-vs.-time data for an aqueous solution of a peptide. The regression lines are for storage temperatures of 80°, 73°, 60°, 50°, 40°, and 30°C. The left-hand panel gives the ln(-slope)-vs.-l/T Arrhenius plot.

The reaction orders obtained from nonlinear analysis are usually nonintegers. It is customary to round the values to nearest integers, half-integers, tenths of integers, etc. as may be appropriate. The regression is then repeated with order(s) specified to obtain a revised value of the rate constant, or revised values of the Arrhenius parameters. 

The variation of the cathodic peak potential with the scan rate (0.3-0.4 mV precision on each determination, 1 mV reproducibility over the whole set of experiments) allows the determination of the rate constant with a relative error of 3-11%. The results are consistent with those derived from anodic-to-cathodic peak current ratios. Simulation of the whole voltammogram confirms the absence of significant systematic errors that could arise from the assumptions underlying the analysis of kinetic data. Activation parameters derived from weighted regression Arrhenius plots of the data points taken at 5 or 6 tern-... 

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