Graphical and Numerical Methods of Analysis

In this chapter you ll learn how to use graphical and numerical methods to solve chemical problems. The methods described range from the simple (finding the roots of a polynomial from a graph of the function) to the complex (using matrix methods to solve sets of simultaneous equations). Excel does most of the work, though, by virtue of its graphing capabilities and built-in numerical analysis capabilities. 

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Because of the very low scattered intensity, the data at the shortest sampling interval is usually the poorest in quality. Arbitrary renormalization of the data followed by the graphical representation outlined above is most likely to amplify errors in the data analysis, focus attention on the inherent errors in the construction of the composite relaxation function, and give undue importance to the worst data. When the data is as limited in quality as it is for this problem, any method of analysis should be as numerically stable as possible and the maximum allowable smoothing of the data should be employed. This procedure may obscure subtle features, but only very high quality data could reliably demonstrate their presence anyway. At the present time a conservative approach seems more sensible. 

The analysis and mathematical modelling of mixtures is significant in pharmaceutical formulation and these present certain particular problems. The final chapters cover methods for optimizing formulations and also treat "mixed" problems where process variables and mixture composition are studied at the same time. In both chapters 9 and 10 we continue to illustrate the graphical and numerical optimization methods described earlier. 

The two major methods used predominantly in the kinetic analysis of isothermal data on solid-catalyzed reactions conducted in plug-flow PBRs are the differential method and the method of initial rates. The integral method is less frequently used either when data are scattered or to avoid numerical or graphical differentiation. Linear and nonlinear regression techniques are widely used in conjunction with these major methods. 

Advanced EMR methods may be used to conduct quantitative measurements of nuclear hyperfine interaction energies, and these data, in turn, may be used as a tool in molecular design because of their direct relation to the frontier orbitals. The Zeeman field dependence of hyperfine spectra enables one to greatly improve the quantitative analysis of hyperfine interaction and assign numeric values to the parametric terms of the spin Hamiltonian. Graphical methods of analysis have been demonstrated that reduce the associated error that comes from a multi-parameter fit of simulations based on an assumed model. The narrow lines inherent to ENDOR and ESEEM enable precise measures of peak position and high-resolution hyperfine analyses on even powder sample materials. In particular, ESEEM can be used to obtain very narrow lines that are distributed at very nearly the zero-field NQI transition frequencies because of a quantum beating process that is associated with... 

The liquid and vapor densities can be determined either by a graphical method, or, preferably, by a numerical least-squares analysis as described in Chapter XXL Both of these methods are described below. 

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