Linear least squares analysis initial rates

S Spreadsheet Summary In the final exercise of Chapter 13 of Appli-"II cations of Microsoft Excel in Analytical Chemistry, the initial-rate method is explored for determining the concentration of an analyte. Initial rates are determined from a linear least-squares analysis and are used to establish a calibration curve and equation. An unknown concentration is determined. 

Hence, linear least-squares analysis of the total pressure dependence of the initial rate of conversion of reactants to products near the inlet of a packed catalytic tubular reactor provides quantitative information about / forward, surf.Rx and Kb via the following procedure ... 

The usual data analysis procedures for the linearizable models typified by Eq. (2) consist of (1) isolating a class of plausible rival models by means of plots of initial reaction-rate data as a function of total pressure, feed composition, conversion, or temperature (2) fitting the models passing the screening requirements of the initial rates by linear least squares, and further rejecting models based upon physical grounds. 

Procedure. The apparatus and general procedures of the capillary IGC experiment are described elsewhere (1.2.36-). Each measurement was conducted at three different earner gas flow rates (between 2 to 20 cm/s). For each experiment, an estimate of a and B was obtained using moment analysis and used as an initial guess for the Fourier domain fitting. The values of B2 at the three different earner gas flow rates were plotted versus 1/tc. Using equation (31), x2/D was estimated from the slope of B2 versus 1/tc, using a linear least-squares. 

All reactions were carried out in a thermostated, magnetically stirred 50 ml autoclave (baffles, two thermocouples, X-shaped stirrer, 1=25 mm, max. 900 rpm). A reservoir and a pressure regulator allowed experiments to be carried out under isobaric conditions, p and T in the autoclave and in the reservoir were recorded every 10 s. Usually, the reactions were stopped after 10 min corresponding to conversions of 10 - 50%. Conversion and e.e. were determined by GLC-analysis of the crude reaction mixture (30 m capillary 3-Dex 100, 75°C). In all experiments with HCd modifier, the R-enantiomer was the major enantiomer. The initial rates of hydrogenation were calculated with a linear least square method from the pressure drop in the reservoir, accounting for the compressibility of hydrogen. In all series, one experiment was carried out without modifier to determine k (see below). 

In nonlinear regression analysis, we search for those parameter t alues that minimize the sum of the squares of the differences beiw een the measured values and the calculated values for all the data points.- Not only can nonlinear regression find the best estimates of parameter values, it can al,so be used to discriminate between different rate law models, such as the Langmutr-Hin-shelw ood models discussed in Chapter 10. Many software programs are available to find these parameter values so that all one has to do is enter the data, The Polymath software will be used to illustrate this technique. In order to carry out the search efficiently, in some cases one has to enter initial estimates of the parameter -alues close to the actual values. These estimates can be obtained using Ihe linear-least-squares technique discussed on the CD-ROM Professional Reference Shelf. 

However, the regression theory requires that the errors be normally distributed around (—7 a). and not around f as in the linearized version just described. Hence use the values determined as initial estimates to obtain more accurate values of the constants by minimizing the sum of squares of the residuals of the rates directly from the raw rate equation by nonlinear least squares analysis. 

Initiation rate constants ( i) for the degradation may be obtained by least-squares analysis of the linear segment of the early portion of this plot. Propagation rate constants ( p) may be obtained in a similar manner fi-om data obtained later in the run when propagation has become the dominant reaction. This is illustrated in Figures 8 and 9. Table 14 shows the rate constants obtained in this manner for the degradation of VDC/butyl acrylate copolymers. 

---