Model easy path

Figure 4.1.5. Easy path model for a two-phase ceramic (a) Schematic representation of grains separated by a discontinuous grain boundary phase, (b) Series circuit equivalent according to Bauerle [1969]. (c) Parallel circuit equivalent according to Schouler [1979].

Figure 4.143. Circuits proposed for modeling the impedance spectrum of polycrystalline sodium )8-alumina (a) Easy path model according to Lilley and Strutt [1979] (b) Multielement model according to De Jonghe [1979].

FIGURE 7-1 Equivalent circuit models representing bulk solution of a two-phase microstructure A. series layer B. parallel layer C. easy path ... 

Hydrotreating reaction models are nicely exemplified by the Amoco Easy-Hard lump model of Figure 5. The global paths and the kinetics reported in this figure were taken from the review by Beaton and Bertolacini (3), which summarizes experimental results, the modelling, and commercialization of Amoco s hydrotreating unit In short, the subdivisions of resid into hard and easy fractions and gas oil into reactant and product fractions were required to account for kinetic nonlinearities. The seven lumps and 14 rate constants provided significant flexibility and were able to describe the kinetics of several relevant product fractions. 

Electron-transfer reactions are also gas-phase processes, which truly correspond to the aims and model examples for reaction dynamics. They have been studied for more than 60 years, and appear as invaluable benchmarks to unravel the interplay between forces during chemical reactions. As a first approximation, electron transfer can be considered as localized and proceeds instantaneously when the reactants move towards each other along the reaction path. The electron transfer leads to a sudden release of forces which drive the further dynamics of the reaction. Hence visualizing and modeling the effect of these forces should be a fairly easy task. 

It is important to note that, in the models chosen for theoretical calculations, the reaction path from cis to the trans form is easy, particularly due to the lack of steric hindrance (R = H) (Scheme 8). If, for example, bulky substituents were linked to the C2 atom, the rotation around the C2-X3 bond would be certainly hindered, increasing the activation barrier of the cis-trans isomerization. In this case, it would be hazardous to take the C-0 bond breaking to be the ratedetermining step in the thermal coloration reaction (Figure 3). It would be possible to observe a ring-opening followed by a ring-closure reaction without any formation of the stable trans colored isomer. 

Model predictions can also be inaccurate due to the incompleteness of the chemical model, e.g. if some reactions or species were incorrectly omitted from the mechanism. If the missing species or reactions are completely missing from the database used by the model-construction software, there is no easy way to detect them (though perhaps a human expert might notice the omission in the tree databases described in Section II). There is certainly chemistry which is not well understood, even in the well-studied thermal gas-phase chemistry of small organic molecules for example some of the important reactions of peroxyl radicals are still unclear (Taatjes, 2006), the true reaction path for CH + N2 was only recently identified (Moskaleva and Lin, 2000), and recently some reactions that occur over ridges rather than saddle points have been identified (Townsend et al., 2004). It will be some time before there is a community consensus on how to correctly generalize from some of these observations. 

An orbital is a volume of space about the nucleus where the probability of finding an electron is high. Unlike orbits that are easy to visualize, orbitals have shapes that do not resemble the circular paths of orbits. In the quantum mechanical model of the hydrogen atom, the energy of the electron is accurately known but its location about the nucleus is not known with certainty at any instant. The three-dimensional volumes that represent the orbitals indicate where an electron will likely be at any instant. This uncertainty in location is a necessity of physics. 

If the design was for a second-order model and examination of the contour plots or canonical analysis (see below) showed that the optimum probably lay well outside the experimental domain, then the direction for exploration would no longer be a straight line, as for the steepest ascent method. In fact, the "direction of steepest ascent" changes continually and lies on a curve called the optimum path. The calculations for determining it are complex, but with a suitable computer program the principle and graphical interpretation become easy. 

The Cole equations are descriptive in their nature. Even so, many have tried to use them for explanatory purposes but usually in vain. If a Cole model all the same is to be used not only for descriptive, but also for explanatory purposes, it is necessary to discuss the relevance of the equivalent circuit components with respect the physical reality that is to be modeled. Because the Cole models are in disagreement with relaxation theory, this is not easy. A more general dispersion model, Eq. 9.43, may help circumvent problems occurring when the characteristic frequency is found to vary and DC paths with independent conductance variables cannot be excluded. 

Synthesis involves a complex set of processes and complex systems often show chaos, in particular in autocatalysis. Chaotic processes diverge two systems that follow the same process but have an unnoticed difference in start concentrations follow completely different paths. Chaos is not recognized if the system is locked into what is called a stable attractor or a limit cycle. Then it seems to be simply causal and is usually easy to model. When the system is in a strange attractor it seems to run amok. Chaos in synthesis can be recognized by several indicators ... 

---