Parabolic behavior

At least four different explanations have been proposed to account for parabolic kinetics. The oldest and best established is the "protective-surface-layer" hypothesis. Correns and von Englehardt (6) proposed that diffusion of dissolved products through a surface layer which thickens with time explains the observed parabolic behavior. Garrels ( 12, 1 3) proposed that this protective surface consists of hydrogen feldspar, feldspar in which hydrogen had replaced alkali and alkaline earth cations. Wollast (j>) suggested that it consists of a secondary aluminous or alumino-silicate precipitate. In either case, a protective surface layer explains parabolic kinetics as follows If the concentration of any dissolved product at the boundary between the fresh feldspar... 

Experimental evidence for long-range electron transfer in polypeptides and proteins had been early accrued.The value of using a metal center as a marker is apparent from the above. The approach can be extended to electron transfer between two proteins which are physiological partners. Metal substitution (e. g. Zn for Fe) can be used to alter the value of AG° and permit photoinduced initiation. The parabolic behavior predicted by (5.86) has been verified for the electron transfer rate constant vs AG° within the adduct between cyt c and cyt bj." ... 

The sensitivity of a to the ET mechanism is particularly useful in revealing deviation from the parabolic behavior expected on the basis of equation (7). This is exemplified by the application of the a convolution analysis to the reduction of a series of perbenzoates in DMF at the glassy carbon electrode. ... 

In our previous studies [2,3], we have shown that for lime-stabilized clayey sand, a parabolic behavior for the stress- strain curve is expeeted. Hence, for our polyamide reinforced materials, a similar behavior can be observed. This is presented in equation 2 for stabilized speeimens containing 15% lime ... 

Because the increase in thickness is expected to follow parabolic behavior, the thicker the product, the slower the growth rate. 

In general the temperature dependence of the nonradiative processes is reasonably well understood. However, the magnitude of the nonradiative rate is not, and cannot be calculated with any accuracy except in the weak-coupling case. The reason for this is that the temperature dependence stems from the phonon statistics which is known. However, the physical processes are not accurately known. Especially the deviation from parabolic behavior in the configurational coordinate diagram (anharmonicity) may influence the nonradiative rate with many powers of ten (11). 

By means of OH- and COOH-containing plasma polymer layers the quahfica-tion of these layers as models of single-type functionalized adhesion promoters with variable concentrations of functional groups should be proved. The plasma-initiated copolymerization of acrylic acid with ethylene or 1,3-butadiene is shown in terms of measured COOH concentration as a function of the composition of the comonomer mixture in Fig. 18.3. Depending on the co-monomer reactivity, a more linear correlation (butadiene), or a parabolic behavior (ethylene), between precursor composition and COOH groups produced was observed. For each type and concentration of functional group, its concentration was determined by chemical derivatization followed by XPS analysis as described in Section 18.2.5. 

As shown in Fig. 18.4, similar tendencies were found for the aUyl alcohol copolymerization with ethylene, butadiene, or styrene. Here, the curves are observed to progress from a parabolic (ethylene) to a nearly linear correlation (butadiene) and to an anti-parabolic behavior (styrene) between measured OH group concentrations and the stoichiometry of the precursor mixture. 

Having this way proved the efficiency of the parabolic energy expression in terms of electronegativity and chemical hardness indices, with the regulatory effects in chemical reactivity principles, the next step consists in discussing their observability character in order to can be employed as viable quanto-computational tools linking the density with many-electronic information and with the energetic parabolic behavior. 

Determination of the Shape of the Band Gap. As the DOS is an average of the metallic islands, the parabolic behavior of valence and conduction bands is not important. Instead, tt and tj bands of the density of states are assumed to be... 

We have performed calculations for t (IOO) with =19 within the (4CA) model by using Eqs. (4.7) and (4-8). For such a number of planes, as one can see from Figs. 10 and 11, the spectra are nearly the same. Notice that in Fig. 11 around the p point, the Rayleigh wave has a parabolic behavior since it corresponds to flexural modes of the slab. Apart this small region the... 

The evolution of the polarizability per unit cell(a22 ) with respect to the chain-length (up to 20 carbon atoms) is depicted in the figure. A change of slope is easily identified in each curve, it corresponds to the limit beyond which the parabolic behavior predicted by the free-electron theory(a, L starts to break down to reach a saturation regime in the fimit of large n. Moreover, the shape of the curves noticeably depends upon the alternation degree. 

Hence, the oxide thickness becomes x = f(T). It is known that the oxide kinetics of many metals and alloys show parabolic behavior. However, other type of behavior is possible and eq. (10.12) is generalized and modeled as an empirical relationship given by... 

If n = 1/2, a parabolic behavior of nonporous, adherent, and protective scale develops by diffusion mechanism. Thus, 1 < PB < 2 and Ae mechanism of sc growth is related to metal cations diffusing through the oxide scale to react with oxygen at the oxide-gas interface. 

Figure 10.12 Parabolic behavior due to weight gain of some oxides listed in Table 10.2. Conditions T = 1000 C and Pq, = 1.0 atm.

Complex rate laws are usually observed for alloys where different oxides are simultaneously formed on the surface leading to changes in the oxidation rate. For example, if at the beginning a more or less nonprotec-tive oxide scale is formed which is undergrown by a protective partial layer then the oxidation rate will initially be higher than later in the oxidation period in other words, sub-parabolic behavior will be measured. Complex rate laws will also follow when gas phase transport through pores or cracks in the scales is involved. This case, however, is not a protective situation and actually represents at least partial failure of a protective scale. 

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