Varieties of curves

Figure 9. Typical stress-strain curve for solid propellants at 0.77 in./min. and 80°F. E is the slope of the tangent to the initial portion of the curve. A variety of curve shapes are possible depending on specific formulations and test conditions...

The quadratic coefficient flu describes the curvature along the Xj axis, and B22 describes the curvature along the X2 axis. The quadratic coefficients can be positive or negative and they can have different numerical values. Quadratic response surface models can therefore describe a variety of curved surfaces allowing analysis of surfaces which are convex and have a maximum or which are concave and have a minimum response as well as surfaces which are saddle-shaped. 

There is a wide variety of curve resolution techniques available [Lawton Sylvestre 1971,Liang etal. 1993, Malinowski 1991,Tauler 1995, Vandeginste et al. 1987, Windig Guilment 1991], In these techniques, constraints can be included to direct the solution to meaningful spectra and concentration estimates. 

The solubility relations of partially miscible liquids have been studied by Guthrie, Alexejefi, Rothmund, Timmermans, Flasch-ner and others. A considerable variety of curves have been obtained, and we shall therefore discuss only a few of the different cases which may be taken as typical of the rest. 

A variety of curves can be generated with this approach for different constants, and the curves can be studied using thermodynamics (see examples in Figures 21.1 and 21.2). Traditionally, thermodynamics deals with the statistical properties of gases and involves quantities such as temperature, pressure, volume, and entropy. Who would guess that quantities such as temperature have recently been used by several authors to describe mathematical curves instead of gases ... 

It is therefore not surprising that equation 31 fits a variety of curve shapes since) with the Langmuir term expsinded to n = 2 (i.e. equation 32), the relation in fact contains the ratio of two quadratics. Nevertheless, and particularly in view of the successful quantitative interpretation of the Langmuir forms of isotherm equations in the instances of the BET model (equations 7, et seq.) and retentions with blended stationary phases in gas chromatography (equations 14, et seq.), the LC relations cannot be dismissed as entirely empirical since, in any event, although such a connection has yet to be established, whatever interpretations are placed on the fitting constants must presumably involve at least the solute activity coefficients in the mobile and stationary phaises (see below) and, most likely, the (finite-concentration) activity coefficients pertinent to the mobile- and stationary-phase components as well. 

In Secdon 14.8, we discussed the concept of curve fitting. MATLAB offers a variety of curve-fitting options. We will use Example 14.11 to show how you can also use MATLAB to obtain an equation that closely fits a set of data points. For Example 14.11 (Revisited), we will use the... 

We first note that Eq. (7.5.1) has enormous flexibility. Depending on the choice for the parameters a, . .., fl4, one obtains a considerable variety of curves displayed in Figure 7.5.1 that you should carefully examine. These curves mimic the ones displayed in Sections 3.8-3.10. Note that the shapes are achieved with the choice = 0. 

In the erosion literature, materials are broadly classified as ductile or brittle, based on the dependence of their erosion rate on a. Ductile materials, such as pure metals, have a maximum erosion rate, E, at low angles of incidence (typically 15 to 30°), while for brittle materials, such as ceramics, the maximum is at or near 90°. These two classical extremes are illustrated in Fig. 9. A variety of curves intermediate between these classical extremes exist and in some cases the same material exhibits behavior that shifts from one extreme to the other, depending on erosion conditions. 

From polarization curves the protectiveness of a passive film in a certain environment can be estimated from the passive current density in figure C2.8.4 which reflects the layer s resistance to ion transport tlirough the film, and chemical dissolution of the film. It is clear that a variety of factors can influence ion transport tlirough the film, such as the film s chemical composition, stmcture, number of grain boundaries and the extent of flaws and pores. The protectiveness and stability of passive films has, for instance, been based on percolation arguments [67, 681, stmctural arguments [69], ion/defect mobility [56, 57] and charge distribution [70, 71]. 

Hi) Surface blockers. Type 1 tlie inliibiting molecules set up a geometrical barrier on tlie surface (mostly by adsorjDtion) such as a variety of ionic organic molecules. The effectiveness is directly related to tlie surface coverage. The effect is a lowering of tlie anodic part of tlie polarization curve witliout changing tlie Tafel slope. 

Type V isotherms of water on carbon display a considerable variety of detail, as may be gathered from the representative examples collected in Fig. 5.14. Hysteresis is invariably present, but in some cases there are well defined loops (Fig. 5.14(b). (t ), (capillary-condensed water. Extreme low-pressure hysteresis, as in Fig. 5.14(c) is very probably due to penetration effects of the kind discussed in Chapter 4. 

The approach that we have worked out for the titration of a monoprotic weak acid with a strong base can be extended to reactions involving multiprotic acids or bases and mixtures of acids or bases. As the complexity of the titration increases, however, the necessary calculations become more time-consuming. Not surprisingly, a variety of algebraic and computer spreadsheet approaches have been described to aid in constructing titration curves. 

An interesting outgrowth of these considerations is the idea that In r versus K or Vj should describe a universal calibration curve in a particular column for random coil polymers. This conclusion is justified by examining Eq. (9.55), in which the product [i ]M is seen to be proportional to (rg ), with r = a(rg 0 ) - This suggests that In rg in the theoretical calibration curve can be replaced by ln[r ]M. The product [r ]M is called the hydrodynamic volume, and Fig. 9.17 shows that the calibration curves for a variety of polymer types merge into a single curve when the product [r ]M, rather than M alone, is used as the basis for the cafibration. 

Direct quantitation of receptor concentrations and dmg—receptor interactions is possible by a variety of techniques, including fluorescence, nmr, and radioligand binding. The last is particularly versatile and has been appHed both to sophisticated receptor quantitation and to dmg screening and discovery protocols (50,51). The use of high specific activity, frequendy pH]- or p lj-labeled, dmgs bound to cmde or purified cellular materials, to whole cells, or to tissue shces, permits the determination not only of dmg—receptor saturation curves, but also of the receptor number, dmg affinity, and association and dissociation kinetics either direcdy or by competition. Complete theoretical and experimental details are available (50,51). 

Steel, copper, and brass fiber may have a variety of aspect ratios, shape, ie, straight versus curved fibers and cross-sectional geometry, surface roughness, and chemical compositions. Fibers having tight specifications in terms of cleanliness, chemical composition, and aspect ratio ate necessary. The fibers are usually machined from larger metallic forms. 

The circular dichroism curves for a variety of penicillanic acid derivatives have been published and discussed (B-77MI51100) and have been used to support extended Hiickel MO calculations (77T711). 

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