Second-order behavior

Sccond-Ordcr Reactions The simplest overall reaction demonstrating second-order behavior is... 

In this chapter the regimes of mechanical response nonlinear elastic compression stress tensors the Hugoniot elastic limit elastic-plastic deformation hydrodynamic flow phase transformation release waves other mechanical aspects of shock propagation first-order and second-order behaviors. 

For many reaction mechanisms, the rate-determining step occurs after one or more faster steps. In such cases the reactants in the early steps may or may not appear in the rate law. Furthermore, the rate law is likely to depart from simple first- or second-order behavior. Fractional orders, negative orders, and overall orders greater than two, all are signals that a fast first step is followed by a slow subsequent step. 

While the reaction exhibits good second-order behavior in 1.83M sulfuric acid, the theoretically linear second-order rate plots develop marked curvature at lower acidities. This observation is illustrated by Figure 3, where the second-order plots are shown for the reaction in 0.950M sulfuric acid, and for two media of lower acidity... 

Thus, for the conditions where second-order behavior is observed, the chemical circumstances indicate the cerium(IV) oxidation of each chromium complex will involve a rate-determining one-equivalent oxidation of the complex ion (or a species in rapid equilibrium with the complex ion) to an intermediate, followed by the rapid one-equivalent oxidation of the intermediate. Without reference to the role of water coordinated to the chromium, the most obvious mechanism in accord with these specifications is ... 

In contrast, geminate recombination, caused, for example, by proximal trapping of the injected electron, would yield power-independent recovery times see Scheme 2. (Of course, if the trapped electrons were not immobilized, but instead were able to migrate rapidly from surface state to surface state, overall second-order behavior would be recovered [37].)... 

Spindler suggested a straightforward modification of the previous theories to produce a model that predicted second-order behavior at low pressures... 

The application of this rate law to the simulation of electrochemical behavior requires two dimensionless input parameters ktf and KC. When these are supplied, three-dimensional chronoamperometric or chronocoulometric working surfaces [34] are generated. These working surfaces both indicate first-order behavior when KC is large and second-order behavior when KC is small. Intermediate values of KC produce the variable reaction orders between one and two that are observed experimentally when the bulk olefin concentration is varied. Appropriate curve fitting of the experimental i(t,C) data to the simulation results in the evaluation of k and K details appear in the referenced work. 

The reaction shows second order kinetics up to [N3 ] of lmM. While calibrating the dead time and performance of the MHQ instrument, the author observed clear deviations from second order behavior at higher [Ns"] including the formation of a short-lived transient intermediate (Figure 4a). 

As this examination shows, the order of a reaction with pathway 5.72 depends on the relative magnitudes of kKA and kK CA. Cases I and II, with first- and second-order behavior, respectively, are the extremes. In between, with the two terms in the denominator of eqn 5.74 of comparable magnitude, the order is between first and second and increases with conversion. It is even possible for the same reactant to have different reaction orders under different conditions, for instance, in different solvents. 

Experimental observations suggest fe2tOx] > fe ][Red] so that 2ki, i.e., simple second-order behavior with Eq. (23), the rate-determining step. Linear relationships, using Eq. (10), suggest that a simple electron abstraction from the diol to give the radical H2Y operates in this rate-determining step. The Marcus equation (Eq. (21)), recasted in a more compressed form of... 

Metal-Ion-Catalyzed Autoxidation. Figure 1 illustrates the variation of the first-order rate constants for the autoxidation of ascorbic acid by molecular oxygen with the concentration of the Cu (II) ion, which is present in catalytic (i.e. low) concentrations (8). The linear relationship indicates second-order behavior [first order in ascorbic acid and first order in Cu(II)]. The catalytic effect of Cu(II) is also seen to decrease... 

H chemical shifts are another matter, since the signals of most protons are multiplets. Strictly speaking, spectral analysis is required to determine the chemical shifts and coupling constants of those protons whose signals display second-order behavior (Chapter 4 and Appendix 4). For those protons whose spectra are first order or even pseudo first order, chemical shifts and couplings can be determined from the spectra by inspection, but some care should be exercised in this endeavor. 

Analyses of the variation of transition temperatures with composition (Fig. 24) reveal an unambiguous departure from linearity (Schofield and Redfern 1993). For example, the plateau region is more extensive than in other oxide systems, ranging from 0 spontaneous strain is plotted as a function of composition (Fig. 25), the data depart slightly from ideal second-order behavior. Schofield et al. (1997) attribute this deviation and the broad plateau to the short-range interaction length of the strain fields associated with the Zn cations. 

Impuritiesand the a P-quartz tranition. The a- 3-quartz transition was the basis for one of the earliest systematic investigations of the variation of transition temperatures in response to impurities. Pure a-quartz undergoes a first-order transition to a microtwinned incommensurate structure at 573°C, and this modulated phase transforms to P-quartz at 574.3°C with second-order behavior (Van Tendeloo et al. 1976, Bachheimer 1980, Dolino 1990). Tuttle (1949) and Keith and Tuttle (1952) investigated 250 quartz crystals and observed that Tc for natural samples varied over a 38°C range. In their examination of synthetic specimens, substitution of Ge for Si raised the critical temperature by as much as 40°C, whereas the coupled exchange of Ar +Li o Si depressed Tc by 120°C. They concluded from their analyses that the departure of the a-P-quartz inversion temperature from 573°C could be used to assess the chemical environ-ment and the growth conditions for natural quartz. 

The first contribution depends upon the second-order behavior of the Hamiltonian operator and the unperturbed reference state wave function, while the second term (which will be subsequently be referred to as the relaxation contribution or relaxation term ) depends on the derivative of the wave function. This is perhaps most easily appreciated by inserting the equation for the wave-function derivative into that for the second derivative of the energy, giving... 

Application of this model to a residuum desulfurization gave a linear relationship. However, it is difficult to accept the desulfurization reaction as a reaction that requires the interaction of two sulfur-containing molecules (as dictated by the second-order kinetics). To accommodate this anomaly, it has been suggested that, as there are many different types of sulfur compounds in residua and each may react at a different rate, the differences in reaction rates offered a reasonable explanation for the apparent second-order behavior. For example, an investigation of the hydrodesulfurization of an Arabian light-atmospheric residuum showed that the overall reaction could not be adequately represented by a first-order relationship. However, the reaction could be represented as the sum of two competing first-order reactions and the rates of desulfurization of the two fractions (the oil fraction and the asphaltene fraction) could be well represented as an overall second-order reaction. 

Establishing the relationship between microscopic and macroscopic second-order behavior of subphthalocyanines still remains a challenging target. Consequently, a number of studies have been performed on SubPc systems in condensed phases. 

In all the experiments, the estimated parameters exhibited an overall second-order behavior for the three reaction rates. The model best fits the experimental data by assuming a first-order rate dependence on the concentration of each component in the rate equation (Eqs. 18-20). The global kinetic constants (Ks, KR,... 

Contrary to these results, Faust et al. (37) found close to second order behavior in [TiCl4] in similar IB polymerizations initiated by aromatic compounds, e.g., dicumyl chloride. The authors explained their findings by assuming the existence of TiCl4 dimers. The resolution of this intriguing discrepancy between these observations must wait for further investigation. 

Figure 11.7 (a) Feedback control (b) the resulting second-order behavior of the liquid level. 

The second term of the right-hand side gives rise to the second-order behavior of the system. Equation (11.31) or its equivalent (11.32) is the starting point for the examples of Appendix 11A. 

The effect of added NO is more straightforward. In a single run under argon, 22 torr of NO was added to the gas cell. The cell contained 5 mL of a methylene chloride solution of pyrene (2.8 X 10 2 M and N(IV) (6.7 X 10 5 M) on a molar basis, [NO] >> [N(IV)]. In this case, the development of product smoothly followed second-order behavior, in contrast with the first-order production of 1-nitropyrene under the same conditions but in the absence of NO. 

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