Curve fitting technique

When an element is present on the surface of a sample in several different oxidation states, the peak characteristic of that element will usually consist of a number of components spaced close together. In such cases, it is desirable to separate the peak into its components so that the various oxidation states can be identified. Curve-fitting techniques can be used to synthesize a spectrum and to determine the number of components under a peak, their positions, and their relative intensities. Each component can be characterized by a number of parameters, including position, shape (Gaussian, Lorentzian, or a combination), height, and width. The various components can be summed up and the synthesized spectrum compared to the experimental spectrum to determine the quality of the fit. Obviously, the synthesized spectrum should closely reproduce the experimental spectrum. Mathematically, the quality of the fit will improve as the number of components in a peak is increased. Therefore, it is important to include in a curve fit only those components whose existence can be supported by additional information. 

Range 1 of the mud pump performance characteristic is defined by the performance of the smallest liner, and range 2 is defined by the remaining liners. The pressure loss in a circulating system, except for bit (p ), can be estimated from numerous theoretical formulas or from a flowrate test. Data obtained from a flowrate test can be approximated using a curve-fitting technique by the following function ... 

FIGURE 11.12 Overextrapolation of data, (a) Nonlinear curve fitting techniques estimate an ordinate maximal asymptote that is nearly 100% beyond the last available data point, (b) The curve fitting procedure estimates a maximal asymptote much closer to the highest available data point. A useful rule is to reject fits that cause an estimated maximal asymptote that is >25% the value of the highest available data point. 

The values of a and P in Eq. (125) are obtained by nonlinear curve-fitting techniques from which the uptake Pe and Ke values as a function of BSA concen-... 

Statistics available in the system include a large set of commonly used analysis techniques, as well as advanced nonlinear curve fitting techniques. Statistical results can be displayed numerically or graphically. 

Using modern computer curve-fitting techniques,102 equations (17) and (27) can be fitted simultaneously with variables X, log CH+ and D, obtaining pATBH+. m, DB and DA as coefficients, which is a more precise technique than calculating log/ values first and using equation (17) and the error function61 discussed above. 

The probit relationship of Equation 2-4 transforms the sigmoid shape of the normal response versus dose curve into a straight line when plotted using a linear probit scale, as shown in Figure 2-10. Standard curve-fitting techniques are used to determine the best-fitting straight line. 

Muj taba, I. M. and S. Macchietto. Efficient Optimization of Batch Distillation with Chemical Reaction Using Polynomial Curve Fitting Techniques. Ind Eng Chem Res 36 2287-2295 (1997). 

The surface complexation models used are only qualitatively correct at the molecular level, even though good quantitative description of titration data and adsorption isotherms and surface charge can be obtained by curve fitting techniques. Titration and adsorption experiments are not sensitive to the detailed structure of the interfacial region (Sposito, 1984) but the equilibrium constants given reflect - in a mean field statistical sense - quantitatively the extent of interaction. 

Figure 11.8. Illustration of linearized digital curve fit technique.

The parameter values on the right-hand side of the equation, are the P values, and were determined by a curve fitting technique (section 4.14). Introducing these values in Eq. (4.71) gives the solid line through the points in Fig. 4.18. 

For example, data obtained by this technique are shown in Table VIII and Fig. 12a for HDS of dibenzothiophene catalyzed by a CoMo/carbon (41). In these experiments, 3,3 -dimethylbiphenyl was used as the biphenyl analog, as the chemistries are identical and only small differences in adsorption behavior are expected. As shown in Table VIII, the ratio of rate constants ((kDo + /chs / hpi) was estimated to be 62. With this ratio held constant, the other rate constants in the HDS pathway matrix were estimated using curve-fitting techniques. 

Quantitative measurements are carried out by use of a single frequency in the conventional i.r. spectrum, but, with digitized spectra and a computer, the entire frequency spectrum of each component in a mixture can be fitted by curve-fitting techniques using such methods as least-squares refinements,5 which can yield an indication as to the precision of the fit. 

Traditional approaches to experimental data processing are largely based on linearization and/or graphical methods. However, this can lead to problems where the model describing the data is inherently nonlinear or where the linearization process introduces data distortion. In this case, nonlinear curve-fitting techniques for experimental data should be applied. 

Van Der Voort17,18 used infrared curve fitting techniques, to show that during the ammoniation of trichlorosilylated silica, the formation of silazanes already starts at 423 K in vacuum conditions and at 523 K in an ammonia atmosphere. Since these temperatures are much lower than the ones provided by Fink, a third mechanism must be included ... 

Mathematically, geometric parameters can be described by using the Fourier Series in polar coordinates (p,9). Thus, given a set of boundary points (x, y) from an object of interest, they can be transformed into the polar coordinates with respect to its geometric center (x, y). A curve fitting technique in polar coordinates can be used to fit this set of points into a Fourier Series such that any point p(0) on this boundary can be expressed by... 

Please verify for yourself that this equation is correct. If one does not know Cmax or cannot make a reasonable guess, then one must use nonlinear curve fitting techniques, the simplest of which is the Solver tool in Excel. 

A typical C Is XPS spectrum obtained at the start of a photolysis series is shown in Fig. 4. Using curve-fitting techniques, the spectrum in Fig. 4 can be decomposed into three main peaks. The small one at 282.5 eV is due to atomic carbon bound to a substrate metal atom (C-M). Some of this intensity is due to residual carbon that remained after sputter-cleaning the surface (<0.03 monolayer) and some is due to carbon produced from butanol photolysis during the time it took to acquire the spectrum. The most intense peak at 284.9 eV is due to alkyl carbons (C-H), while the peak at 286.5 eV is due to the chiral C atom bound to the OH group [123]. 

While convenient, it should be noted that most of these approximations require the same precision of measurement as do curve fitting techniques and obviously lead to less accurate results. They have been particularly used to derive values for high melting temperature metals and alloys, yielding values that have an author to author reproducibility of typically 10%, as demonstrated by examination of collected data for Cu and Fe, (lida and Guthrie 1988). It is, however, often difficult when making such comparisons to decide whether the differences reflect varying precisions of measurement, differences in experimental procedure or qualities of materials. 

Several graphical curve-fitting techniques have been developed (see Padday [53] for details) that can be used in conjunction with the numerical integration of the Laplace equation by Bashforth and Adams (and by subsequent workers) to determine d and to obtain y v. Smolders [54,55] used a number of coordinate points of the profile of the drop for curve fitting. If the surface tension of the liquid is known and if 0 > 90, a perturbation solution of the Laplace equation derived by Ehrlich [56] can be used to determine the contact angle, provided the drop is not far from spherical. Input data are the maximum radius of the drop and the radius at the plane of contact of the drop with the solid surface. The accuracy of this calculation does not depend critically on the accuracy of the interfacial tension. 

In this test, we used the thermodynamic equation previously described, and, by curve fitting techniques, judge the accuracy of the implied fit of the experimental data versus the data generated from the proposed equation. This leads us to the consideration of the parameters and equations given below ... 

As can be noted, the correlation between the calculated and actual results, using a very simple curve fitting technique, yields good fits concerning molar volume, which is examined in terms of density, and in cloud point data, examined in terms of tai jerature. For those not familiar with the cloud point phenomena, nonionic surfactants exhibit inverse solubility characteristics per taiperature increases, i.e., as the... 

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