First-order rate constants linear least squares

Data from tests at 250,275,300, and 325 C were used to calculate pseudo-first order rate constants for the formation of H2S. These data are expressed on a standard Arriienius plot (Fig. 2) for which the linear least squares coefficient of determination, r, is 0.98. The apparent activation energy calculated from the slope is 28.5 kcal/mol. This result is in excellent agreement with the recent work of Abotsi, who studied the performance of carbon-supported hydrodesulfurization catalysts (10). Using Ambersorb XE-348 carbon lo ed with sulfided ammonium molybdate (3% Mo loading) prepared by the same procedure reported here, Abotsi hydrotreated a coal-derived recycle solvent The apparent activation energy for... 

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. 

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. 

The functional dependency of the rate constant on temperature is examined in Figure 24. The first-order rate constant is plotted as a function of Cto/Cpo at 130,150, 170 and 200°C, and linear fits to the data are shown. The best-fit lines were found to pass very close to the origin accordingly, fits were then performed which "forced" each line to pass through the origin. In all of these cases, excellent correlation with the data was maintained, and the results are summarized in Table 3. Here q i is the slope of the least squares fine and r is its correlation coefficient. 

Figure 8. Reduction of nitrobenzene (NB) in aqueous suspensions containing 200 mgL magnetite and 1.5 mM Fe at pH 7. Plot of ln(C/Co) versus time. C and Cq are the concentrations of NB at time zero and t, respectively. The initial pseudo-first order rate constant, kobs, is obtained by a linear least-squares fit of ln(C/Co) = - kobs t, using only the first few data points. Data from (38).

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. 

If the reaction order (n) with respect to component A is known in advance, the reaction model in Equation 8.14 can be integrated. Assuming the reaction is first order in component A ( n= 1), the rate constant, k, can be determined by the non-linear least-squares optimisation indicated in Equation 8.16 ... 

The following example illustrates the ease with which the Solver can be used to perform non-linear least-squares curve fitting. Here we analyze kinetics data (absorbance vs. time) from a biphasic reaction involving two consecutive first-order reactions (A =— B =— C) to obtain two rate constants and the molar absorptivity of the intermediate species B. 

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 ... 

Rates of hydrolysis were determined spectrophotometrically at 37.5 C using a Beckman DU-2 instrument. An aliquot of a standard solution of the metal salt was added to 3 mL of a 0.5 x 10 molar solution of PVA-QA in 50% aq. ethanol buffered with 0.05M HEPES at pH 7.5 in a standard cuvette, and the change in absorbance as a function of the time was measured for at least 3 half-lives at the following wavelengths Ni(II) acetate 287 nm. Co(II) acetate 277 nm, Cu(II) nitrate 286 nm, Zn II) sulfate 280 nm. The rates of hydrolysis of PVA-QA films were measured at 37.5°C in water buffered with 0.05M HEPES at pH 7.5. The rate constants were derived from the kinetic data by the method of Swain et al. or by a non-linear least square curve fitting program capable of deconvoluting consecutive and simultaneous first order processes. ... 

Using linear least squares, determine whether the reaction obeys first-order, second-order, or third-order kinetics and find the value of the rate constant. 

Assuming that the reverse reaction is negligible, determine whether the reaction is first, second, or third order, and find the value of the rate constant at this temperature. Proceed by graphing ln(c), 1 /c, and 1 /c, or by making linear least-squares fits to these functions. Express the rate constant in terms of partial pressure instead of concentration. 

For a first-order reaction, a plot of In [A] or ln(PP, — PPjnf) against time will be linear with a slope of k. This approach is still in use for the determination of rate constants, but it is being rapidly replaced by least-squares fitting of the data to rate equation 8.7 or 8.9. This faster and more precise data treatment has become widely available through the accessibility of computers and appropriate software. 

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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