Treat Experimental Data

Sum of Growth Rate Distributions. Berglund and de Jong (1990) devised a method of data treatment that allows use of the entire size distribution. It was previously shown that the moments of the growth rate distribution can be related to the moments of the size distribution, thus. 

From this relation the growth rate distribution that causes curvature can be determined by guessing different forms and checking against the mean sizes. 

Upon successful fit of data the nucleation and growth kinetics can be determined. 

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Until now we have looked at various aspects of light scattering under several limiting conditions, specifically, C2 = 0, 0 = 0, or both. Actual measurements, however, are made at finite values of both C2 and 6. In the next section we shall consider a method of treating experimental data that consolidates all of the various extrapolations into one graphical procedure. 

A kinetic model based on the Flory principle is referred to as the ideal model. Up to now this model by virtue of its simplicity, has been widely used to treat experimental data and to carry out engineering calculations when designing advanced polymer materials. However, strong experimental evidence for the violation of the Flory principle is currently available from the study of a number of processes of the synthesis and chemical modification of polymers. Possible reasons for such a violation may be connected with either chemical or physical factors. The first has been scrutinized both theoretically and experimentally, but this is not the case for the second among which are thermodynamic and diffusion factors. In this review we by no means pretend to cover all theoretical works in which these factors have been taken into account at the stage of formulating physicochemical models of the process... 

Numerical methods have been used in solvent extraction chemistry for treating experimental data for many years. As shown in Chapter 4 (see section 4.14.3), modeling of extraction processes in terms of a set of assumed equilibria with adjustable (best-fit) coefficients was widely used to identify and characterize the major species formed in these processes. The improvement of this approach, directed toward studying systems of greater diversity and complexity, including corrections for thermodynamic activity... 

Due to the large number of various types of slow relaxations irreducible to each other, in treating experimental data one must establish precisely what relaxation time is high. 

If chaotic dynamics are present, the experimental errors do not originate exclusively from classical randomness. Thus, the measures of central tendency used to describe or treat experimental data are questionable, since averaging is inappropriate and masks important information in chaotic systems [234]. 

Tidwell and Mortimer noted that the Fineman and Ross method does not treat experimental data with equal weight, yielding physically unrealistic results in some cases [172]. In order to remove this drawback, Tud6s et al. [173] proposed a modification of eqn. (80) which can also be evaluated graphically... 

In the experimental works discussed in this book, the real values of asu = E /k T have ranged from about 15 to 30. If so, just the first terms of the above power series 2.30 or 2.31 suffice to provide the precision that is required when treating experimental data. It can be verified that the series truncated this way yield a common approximate formula for the six integrals ... 

When treating experimental data, calculation of the moments is straightforward. Hence, the above formulae offer a convenient way to fitting the shapes. 

The approach described above has been applied to treat experimental data on adsorption of pyridine from the electrolyte solutions [61]. Using Eq. 28 made it possible to determine the slip length as a function of surface excesses of pyridine. In agreement with the theoretical prediction, it was found that bs grows with Ta. The values of bs did not exceed 0.3 nm and 1.2 nm for adsorption from butanol and water solutions, respectively. The dependence of slip length on surface excess was essentially linear (Eq. 30) for pyridine adsorption from butanol solution, but deviated from linearity for pyridine adsorption from water. The deviation was attributed to a reorientation of adsorbed pyridine molecules at the Au surface. 

In Section 2.2.2 we have simulated the effects of various complicating events by computer generated diagnostic plots. In this section we show how to treat experimental data to obtain kinetic parameters in the presence of the following... 

The Temkin and Freundlich adsorption isotherms have been sucessfully applied to treat experimental data not only for adsorption of gases (hydrogen) on metal surfaces and ions on metal electrodes, as well as the adsorption of proteins to functional surfaces. The explicit expression for reaction kinetics in the case of biographic (intrinsic) nonuniform surfaces can be derived only in a few cases, which in fact limits the possibility of wider application of the a priori nonuniform surfaces. 

There are also other techniques for treating experimental data, and other specific experiments possible, than the simple methods discussed here. Measurements of reaction half-life, as defined in equations (1-45), are directly related to rate constants and also provide a simple means for determination of reaction... 

Deriving a theoretical expression for p/Capp requires evaluation of an electrostatic model for the polyelectrolyte chain, a procedure with significant uncertainties and much analytical and/or computational difficulty. Consequently, two empirical approaches are often employed to treat experimental data. In the first, one considers a modified Henderson-Hasselbalch equation, as for example, the form shown below which includes a single empirical parameter n (187,191) ... 

McKetta and co-workers have discussed various graphical and analytical methods of treating experimental data. Butcher and Dadson and Canfield, Leland, and Kobayashi have described graphical methods that yield more accurate values of N. The latter method has been used recently by Lange and Stein. A computational method which extracts the virial coefficients from the experimental data by iteration has been described by Hoover et al. ... 

Obviously, the way of considering the transmission function must be consistent with that used to treat experimental data. Here the influence of the photoemission anisotropy parameter (L) is dropped, which is strictly correct when s peaks are used or y is close to the magic angle (cf. the section on Angular Asymmetry Factor). 

Consequently, the curve of a rate vs. a concentratiOT is linearized on rp — coordinates (Lineweaver-Burk coordinates) or on rp — rpJCs coordinates. The both types of coordinates can be used when treating experimental data for solving the inverse kinetic problem. The method of initial rates, when initial reaction rates are measured at different initial substrate concentrations, is used for this purpose. 

Equation (38) has been employed for treating experimental data in [22, 26-28]. If the one-phonon function f( v) found from PSB by means of Eq. (38) is correct, the Debye-Waller factor calculated with the help of f(v) by means of Eqs. (19) and (28) must describe the temperature dependence a(T) found experimentally. This verification of the function f(v) has been realized in [22] for perylene doped n-heptane. Results are shown in Fig. 6b. Good agreement of... 

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