Curve-Fitting Method

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). 

In a curve-fitting method the concentration of a reactant or product is monitored continuously as a function of time, and a regression analysis is used to fit an appropriate differential or integral rate equation to the data. Eor example, the initial concentration of analyte for a pseudo-first-order reaction, in which the concentration of a product is followed as a function of time, can be determined by fitting a rearranged form of equation 13.12... 

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. ... 

Equation 13.23 can also be used as the basis for a curve-fitting method. As shown in Figure 13.14, a plot of In(Ct) as a function of time consists of two regions. At short times the plot is curved since A and B are reacting simultaneously. At later times, however, the concentration of the faster-reacting component. A, decreases to 0, and equation 13.23 simplifies to... 

The IC50 can thus be accurately determined by fitting the concentration-response data to Equation (5.1) through nonlinear curve-fitting methods. Some investigators prefer to plot data in terms of % inhibition rather than fractional activity. Using the mass-balance relationships discussed above, we can easily recast Equation (5.1) as follows ... 

Stability constant too large to be reliably calculated using curve-fitting method. No evidence of binding was seen. 

Results of these studies are very encouraging and Indicate that a reasonably fast, accurate, and practical method for the quantitative determination of minerals In complex solids can be achieved with this approach, particularly If multivariate least squares curve fitting methods can be automated. 

After rearranging Eq. (2), the values of ixDCD and KB can be estimated by nonlinear least squares curve-fitting methods (similar to the Michaelis-Menten equation) or the expression can be rearranged under different linear forms (y = rnx I n), where y = (jueff yiD) and x = [CD], Well known are... 

Here we assume that [H+] from acidic or basic impurities that may influence the dissociation equilibrium. If the values of Ka, Kf(HAi), Qand Cs are known, we can get a(H+) by solving the quadratic equation. Figure 3.3 shows the simulated titration curves for various values of Kf(HAy)(Ca+Cs). If we compare the experimental titration curve with the simulated ones (curve-fitting method), we can get the value of K HA ) (see Fig. 6.6). The value of Kf(HAy) can also be obtained by the relation ... 

In order to estimate proper n and Bg(n ) values which satisfy the experimental data, a curve fitting method was employed. 

The use of metal ions at trancer concentrations, if possible by employing radioactive tracers, enables one to carry out such extractions where CM < [HA] or [B], thereby permitting the approximation, that the equilibrium concentrations of either extractants are the same as their initial concentrations. Besides, use of radioactive tracers makes the distribution measurements convenient and fast. Such a condition makes the quantitative interpretation of the extraction data much simpler when the slope-analysis, Job s and the curve-fitting methods, described below, are employed. 

Another factor which influences the speed in performing an analysis is calibration of the instrument. Calibration is especially time-consuming in cases where different elements are run on every analysis but even in cases where the same elements are determined time after time, the frequency of instrument calibration required to maintain a desired level of accuracy is an important consideration. Since manual data collection is not feasible in multielement determinations, the ideal system would undoubtedly be computerized. The computer would handle all data collection steps, the construction of calibration curves by mathematical curve-fitting methods, and the calculation of concentrations from these curves. 

Furthermore, the time to arrival (TTA) can be calculated as time difference between appearance of the first contrast agent molecules in the tissue of interest and contrast agent injection. When using the curve fitting method, TTA equals TA. 

The analysis of spectral shape of XANES not only for Al, but also for other absorbing atoms using a curve fitting method will be therefore helpful to understanding not only the search of structural transition process, but also chemical species appearing in the process. 

Standard Potential (Em0 Volts) of Ag—AgCl Electrode (Cell I) Computed by the Curve-Fitting Method... 

Simplex Curve-Fitting Method and by Arrhenius Method. 

The EXAFS data were analyzed to obtain the r and N values by using a one-shell curve fitting method except for acetic acid and f-butyl alcohol solutions. The results are plotted against the. <4 values in Figs. 3 and 4. The r values pertain to the distances of... 

Two types of secondary structure analysis were used on the spectra collected in this study. The classic curve-fitting method (13) for analysis of the amide I band, was performed in two stages. The first step in the analysis is band narrowing, which allows visualization of component bands, using derivatization and Fourier self-deconvolution (FSD) (14). 

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