Parametric equation

Construction of Alignment Charts. Of the ways to constmct alignment charts, the bmte force method, which requires some idea of the geometry for the chart, is the easiest method to use. The mathematical method, which uses parametric equations of scale to determine the placement and scale of each axis, is the most accurate, but the most difficult to apply. 

Each of the parametric equations that can be formed from an expression represents an axis on the chart. Each set of parametric equations must simultaneously agree with the equation they represent. In other words, on a line drawn through any two variables, a third variable can be found which satisfies the parametric equations. So an evaluation of the chart requites that values produced by each parametric equation be on the chart as a line. A determinant can be used to determine whether or not points are collinear. The parametric equations must be evaluated so they always produce values which he on a line. By replacing the x andjy points with parametric equations of scale for the chart, it is possible to create any diagram. This method can be used to determine the placement of the axes, because the parametric equations can be transformed into equations of scale. 

Parametric Equations It is frequently useful to write the equations of a curve in terms of an auxiliary variable called a parameter. For example, a circle of radius a, center at (0, 0), can be written in the equivalent form x = a cos < >, y = a sin < > where 0 is the parameter. 

The choice of the momenta, rather than of the forces, is illustrated by the simple example of a two-dimensional harmonic oscillator described by the parametric equations... 

It is seen from this comparison that the geometrical parameters obtained by the IPM and by the density functional theory (DFT) methods are close. It is enough to multiply re by 1.44 to obtain the same C H O distance for the TS calculated by the DFT method. The following parametric equations were proposed for the estimation of the C—H, O—H, and C—O distances in the reaction center of TS for reactions ROO + RH in the IPM method [34,35],... 

When inhibited oxidation is quasistationary with respect to hydroperoxide, the induction period t can be expressed through [InH]0, [RH], v , and the rate constants of key reactions (see Equations [8.8] [8.14]). Parametric equations make it possible to derive simple expressions for t. Table 14.8 summarizes expressions for log x in terms of k2 and k2, T, /, and /3 = k2/kd. 

However, the factors of electrophilic solvation and unexpected molar volume have the little influence too. The dependence of IgQ on the solvent property can be satisfactory described by the two-parametric equation too and... 

In this respect, the solvatochromic approach developed by Kamlet, Taft and coworkers38 which defines four parameters n. a, ji and <5 (with the addition of others when the need arose), to evaluate the different solvent effects, was highly successful in describing the solvent effects on the rates of reactions, as well as in NMR chemical shifts, IR, UV and fluorescence spectra, sol vent-water partition coefficients etc.38. In addition to the polarity/polarizability of the solvent, measured by the solvatochromic parameter ir, the aptitude to donate a hydrogen atom to form a hydrogen bond, measured by a, or its tendency to provide a pair of electrons to such a bond, /, and the cavity effect (or Hildebrand solubility parameter), S, are integrated in a multi-parametric equation to rationalize the solvent effects. 

Eq.(18) has two complex roots with real part equal to zero, and consequently it is possible to deduce a relation between x and y. By substituting Eq.(18) into Eq.(12) one obtains a parametric equation xo = fi y )- Eliminating xo between xo = fi y ) and Eq.(13), the parametric equations of self-oscillating behavior are deduced ... 

Exercise 5. Taking into account exercise 4 prove that the parametric equations of curve with cusp point are the following ... 

The enthalpies of dilution of BE were required to calculate the enthalpies of transfer (19). From these Integral enthalpies of dilution AHj.jj the relative apparent molar enthalpies were derived following the technique of Fortier et al (21). The values of AHjjj corresponding to the Initial and final molalities are given In Table 1 along with the parametric equation for. ... 

Unlike the curves you may have seen in geometry books (such as bullet-shaped paraboloids and saddle surfaces) that are simple functions of x and y, certain surfaces occupying three dimensions can be expressed by parametric equations of the form x = f(u,v), y = g(u,v), z = h(u,v). This means that the position of a point in the third dimension is determined by three separate formulas. Because g, and h can be anything you like, the remarkable panoply of art forms made possible by plotting these surfaces is quite large. For simplicity, you can plot projections of these surfaces in the x-y plane simply by plotting (x,y) as you iterate u and in a... 

Considering 8 as variable, this can be read off as the parametric equation of a unit circle, whose Cartesian coordinates are given by the usual projections ... 

It should be noted that the above-mentioned LFER and parametric equations have been derived for a relatively small series of ligands and metals. Therefore, their application for prediction is limited to the class of metals or ligands used for fitting the empirical parameters of those equations. Nevertheless, they have contributed to the fundamental understanding of complexation phenomena, especially for the classes of ligands and metals studied. 

It can be shown that the paramagnetic Curie temperature, at which M vanishes, may be obtained from the vector-valued function M(H), when that function is written as two parametric equations (the parameter being a of Eq. 8.28). The solution is obtained graphically (see, for example, Goodenough, 1966, p. 81) and is found to be ... 

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