Direct Curve Fitting

An estimate of ri and T2 can be obtained from the slope of the experimental F versus /i plot by comparison with curves based on Eq. (7.17) to choose, by trial and error, the values of ri and ri for which the theoretical curve best ts the data. A limitation of this method is the relative insensitivity of the curves to small changes in ri and ri. Another limitation is the assumption imphed in using the differential form of the copolymerization equation [Eq. (7.11) or (7.17)] that the feed composition does not change during the experiment, which is obviously not true. To minimize the error, the polymerization is usually carried out to as low a conversion as possible at which a suf dent amount of the copolymer can still be obtained for direct analysis. The aforesaid limitations can be overcome, however, by the use of an integrated form of the copolymer composition equation, such as Eq. (7.23). In one method, for example, one determines by computational techniques the best values of ri and rz that t Eq. (7.23) to the experimental curve of f or /2 versus (1 - N N ). 

Some representative values of r and ri in radical copolymerization for a number of monomer pairs are shown in Table 7.1. These are seen to differ widely. The reactivity ratios obtained in anionic and cationic copolymerizations are given and discussed in Chapter 8. 

Note Though only single values are shown for ri and r2, the experimentally reported reaetivity ratios often span some range (with a factor of 2 or much larger variation) due to several sources of experimental and statistical uncertainties. 

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Even with the Kelen-Tudos refinement there are statistical limitations inherent in the linearization method. It has been shown [18] that the independent variable in any form of the linear equation is not really independent while the dependent variable does not have a constant variance. The most statistically sound method of analyzing the experimental composition data is the nonlinear method which involves direct curve fitting to the copolymer composition equation. 

Direct curve fitting of the EXAFS data in k-space... 

In the first pari of this project, the analytical form of the functional relationship is not used because it is not known. Integration is carried out directly on the experimental data themselves, necessitating a rather different approach to the programming of Simpson s method. In the second part of the project, a curve fitting program (TableCurve, Appendix A) is introduced. TableCurve presents the area under the fitted curve along with the curve itself. 

Curve-Fitting Methods In the direct-computation methods discussed earlier, the analyte s concentration is determined by solving the appropriate rate equation at one or two discrete times. The relationship between the analyte s concentration and the measured response is a function of the rate constant, which must be measured in a separate experiment. This may be accomplished using a single external standard (as in Example 13.2) or with a calibration curve (as in Example 13.4). 

Miscellaneous Methods At the beginning of this section we noted that kinetic methods are susceptible to significant errors when experimental variables affecting the reaction s rate are difficult to control. Many variables, such as temperature, can be controlled with proper instrumentation. Other variables, such as interferents in the sample matrix, are more difficult to control and may lead to significant errors. Although not discussed in this text, direct-computation and curve-fitting methods have been developed that compensate for these sources of error. ... 

Experimental data that are most easily obtained are of (C, t), (p, t), (/ t), or (C, T, t). Values of the rate are obtainable directly from measurements on a continuous stirred tank reactor (CSTR), or they may be obtained from (C, t) data by numerical means, usually by first curve fitting and then differentiating. When other properties are measured to follow the course of reaction—say, conductivity—those measurements are best converted to concentrations before kinetic analysis is started. 

Kinetic data from Table 2-5 for the reaction between Puvl and Ulv. The time lag plot, right, was constructed according to Eq. (2-42) with r = 300 s. The direct plot according to Eq. (2-33) shows the curve fitted by nonlinear least squares. 

One point, which is often disregarded when nsing AFM, is that accurate cantilever stiffness calibration is essential, in order to calculate accurate pull-off forces from measured displacements. Althongh many researchers take values quoted by cantilever manufacturers, which are usually calculated from approximate dimensions, more accurate methods include direct measurement with known springs [31], thermal resonant frequency curve fitting [32], temporary addition of known masses [33], and finite element analysis [34]. 

B0 and B, are the amounts bound initially (at 1 = 0) and at specific times (t) after initiating dissociation. A plot of log,/l, against l is linear with a slope of -k, k may thus be estimated directly from the slope of this plot or may be obtained by nonlinear least-squares curve fitting to Eq. (5.12). It is always desirable to plot log,/) , against l to detect any nonlinearity that might reflect either the presence of multiple binding sites or the existence of more than one occupied state of the receptor. 

This expression can be modified to apply directly to any of various techniques used to measure the interaction parameter, including membrane and vapor osmometry, freezing point depression, light scattering, viscometry, and inverse gas chromatography [89], A polynomial curve fit is typically used for the concentration dependence of %, while the temperature dependence can usually be fit over a limited temperature range to the form [47]... 

The least-squares method is also widely applied to curve fitting in phase-modulation fluorometry the main difference with data analysis in pulse fluorometry is that no deconvolution is required curve fitting is indeed performed in the frequency domain, i.e. directly using the variations of the phase shift and the modulation ratio M as functions of the modulation frequency. Phase data and modulation data can be analyzed separately or simultaneously. In the latter case the reduced chi squared is given by... 

Equation (8a) lists the cumulative fractions observed at given time points, e.g., at 10, 20, 60 min these are directly given by the raw data. Equation (8b) records the times to reach specified fractions, e.g., 20%, 60%, 80% these must be computed by interpolation or curve fitting. 

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