Linear least squares analysis first-order rate constants

If this procedure is followed, then a reaction order will be obtained which is not masked by the effects of the error distribution of the dependent variables If the transformation achieves the four qualities (a-d) listed at the first of this section, an unweighted linear least-squares analysis may be used rigorously. The reaction order, a = X + 1, and the transformed forward rate constant, B, possess all of the desirable properties of maximum likelihood estimates. Finally, the equivalent of the likelihood function can be represented b the plot of the transformed sum of squares versus the reaction order. This provides not only a reliable confidence interval on the reaction order, but also the entire sum-of-squares curve as a function of the reaction order. Then, for example, one could readily determine whether any previously postulated reaction order can be reconciled with the available data. 

Degradation rate constants were obtained by linear regression least squares analysis of plots of log % EDB remaining vs time. Pseudo-first order rate constants were used to generate Arrhenius plots (log rate constant vs 1/T °K) to estimate activation energies (E ) and to make extrapolated estimates of rate constants and half-life values at ambient temperature. 

Step 1. Use the integral form of a linear-least squares analysis to determine the best value of the pseudo-first-order kinetic rate constant, i, that will linearize the reaction term in the mass transfer equation. It is necessary to apply the Leibnitz rule for differentiating a one-dimensional integral with constant limits to the following expression ... 

Half-Life Dependence of Conventional Pseudo-First-Order Rate Constants. The procedure involves converting absorbance—time profiles (Abs—t) to —In (1 — E.R.)—time profiles which is the traditional way to obtain a first-order rate constant. The —In (1 — E.R.)-time profiles are then subjected to linear least squares analysis on five different segments of the data over time ranges corresponding to 0—0.5 half-Hves (HL, when E.R. =0.5), 0-1 HL, 0-2 HL, 0-3 HL, and 0-4HL. The response of the single-step mechanism to this procedure is that the five apparent pseudo-first-order rate constants are equal within experimental error. This is a very simple, but effective method to differentiate single-step and complex reaction kinetics. 

Both the linear and nonlinear least-squares analyses presented above assume that the variance is constant throughout the range of the measured variables. If this is not the case, a weighted least-squares analysis must be used to obtain better estimates of the rate law parameters. If the error in measurement is at a fixed level, the relative error in the dependent variable will increase as the independent variable increases (decreases). For example, in a first-order decay reaction (Ca = if the error in concentration measurement is... 

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