Taking Derivatives of Experimental Data

Taking derivatives of experimental data (i.e. for determining the coefficient of linear thermal expansion) is not quite as straightforward as taking derivatives of algebraic functions, since data tend to have some scatter. If, for example, a data set has a visually upward trend but two adjacent points are stacked on top of each other, the slope between these points is infinite. An improvement would be to average the slopes from a cluster of points, but if infinity is one of the values, the average value is still infinity. 

INPUT Enter output (derivative) filename , fileout  

OPEN filein FOR INPUT AS 1 OPEN fileout FOR OUTPUT AS 2 

INPUT Enter number of points to be averaged , ptsX IF pts /, / 2 INT(pts /, / 2) THEN PRINT Value needs to be an odd number  

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Figure 4.3 Five point slope calculation for taking derivatives of experimental data.

Differentiating the R x) profile gives the extension rate down the fiber. The practice of taking derivatives of experimental data is prone to errors. Fitting the data first with a spline function can improve accuracy of dR/dx (Secor, 1988). 

Taking the derivative of experimental data is usually a rather inaccurate procedure. 

The process of taking the derivative of experimental data often leads to considerable uncertainty in the dispersion parameter. 

This equation directly relates the relative intensities Irel(i) to the absolute values of the partial Auger rates PA(i), provided PA(all K-LL), or equivalently TA(all K-LL), is known. Following [MSV68, Meh85], this latter quantity can be derived from experimental data as follows T(ls) is known to be T(ls) = 0.27(2) eV (Table 2.2). Taking into account the different branches for the decay of this Is hole-state, one has (see equ. (2.25d))... 

The key concept of the analysis developed here is the interaction coefficient, which we will use to assess the net interactions (favorable or unfavorable) taking place between ions and an RNA. We first introduce interaction coefficients by describing the way they might be measured in an equilibrium dialysis experiment, and give an overview of their significance. These parameters are defined in more formal thermodynamic terms in Section 2.2 and are subsequently used to derive formulas useful in the interpretation of experimental data. 

Values of the ionic radii are derived from experimental data, which give the internuclear distances and electron densities, and generally take the distance of contacting neighbor ions to be the sum of the ionic radii of the cation and anion ... 

In discussing the theory of Debve and Huckel (1923) we shall skip entirely the derivation of the final results, on the basis that these steps transcend the methodology of classical thermodynamics. As a purist one is forced to take the view that the expressions listed below represent excellent limiting laws that are known empirically to represent a very large body of experimental data. This obviously obscures the fact that a detailed understanding is available on the basis of statistical thermodynamics and electrodynamics for dealing with ionic interactions in solutions. 

The thickness of the transition layers for the same adsorbents is given in Table 2 and compared with the values derived from experimental data of Schmidt for the same type of material. It should be noted that for modified silica samples 6 values decrease as the length of the hydrocarbon chain decreases. There exists good consistency of these values derived from various experiments. In the case of RP-2 silica the 6 value seems to be overestimated, if we take into account that silanizing reagent is dimethyl organic phase. 

The disadvantages are, of course, that the derived results depend on the severity of the NDO scheme and the values of the parameters. The latter depend on the quality and availability of experimental data. One seeks to derive a single set of atomic parameters which is capable of handling any environment that the atom finds itself in. The models are invariably valence AO only which therefore provides only limited flexibility. For example, the + 3 and + 5 oxidation states of phosphorous are distinct and place quite different demands on a model which seeks to describe both cases with a single set of valence 3s and 3p orbitals. One must take care to include representative examples of molecules with both oxidation states in order to derive a good set of parameters which will treat both equally. 

Paruta-Tuarez et al. (2011) analyzed the Princen and Kiss equation (Princen and Kiss 1986) associated with the linear form of the function ( >v), (E(Oy) = A(Oy — Oc)). This function was proposed to take into account the experimental dependence of storage modulus (G ) on the dispersed-phase volume fraction (Oy). However, it was found that the Princen and Kiss equation underestimates the storage modulus values in some cases, due to the particular set of experimental data used for derivation of ( >y). Thus, despite the applicability of the linear form of the functicMi ( y) as proposed by Princen and Kiss, it is not universal and another choice of experimental data could lead to other mathematical functions (Paruta-Tuarez et al. 2011). 

The kinetic equation (2.151) came from a theoretical derivation of experimental results. The relation between the reaction rates of various elementary steps still needs to make verification by new experimental data. With the formation and development of chemical reaction engineering, one takes care not only for the micro-mechanism of catal3dic reactions, but also for what kinetic model and its parameters can more properly fit, correlate and predict the dynamic behavior of a catalyst in industrial reactor. Therefore, a new research method by macrosimulation-mathematics modeling is developed to fulfill the requirement of reaction engineering. 

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