Simplification facts

The course instructors or professors have not tried to intentionally deceive their students. Most individuals cannot grasp the full depth and detail of any chemical concept the first time that it is presented to them. It has been found that most people learn complex subjects best when first given a basic description of the concepts and then left to develop a more detailed understanding over time. Despite the best elforts of educators, a few misconceptions are at times possibly introduced in the attempt to simplify complex material for freshmen students. The part of this process that perpetuates any confusion is the fact that texts and instructors alike often do not acknowledge the simplifications being presented. 

Another conventional simplification is replacing the whole vibration spectrum by a single harmonic vibration with an effective frequency co. In doing so one has to leave the reversibility problem out of consideration. It is again the model of an active oscillator mentioned in section 2.2 and, in fact, it is friction in the active mode that renders the transition irreversible. Such an approach leads to the well known Kubo-Toyozawa problem [Kubo and Toyozava 1955], in which the Franck-Condon factor FC depends on two parameters, the order of multiphonon process N and the coupling parameter S... 

Acoustic and similarity methods provide useful information in relation to the mechanism of blast generation by gas explosions. These methods of solution, however, require drastic simplifications such as, for instance, symmetry and constant flame speed. Consequently, they describe only hypothetical problems. In point of fact, because of a complex of flame-flow interactions, freely propagating flames do not have constant flame speeds. Furthermore, these methods do not cover decay characteristics. 

Calculational problems with the Runge-Kutta technique also surface if the reaction scheme consists of a large number of steps. The number of terms in the rate expression then grows enormously, and for such systems an exact solution appears to be mathematically impossible. One approach is to obtain a solution by an approximation such as the steady-state method. If the investigator can establish that such simplifications are valid, then the problem has been made tractable because the concentrations of certain intermediates can be expressed as the solution of algebraic equations, rather than differential equations. On the other hand, the fact that an approximate solution is simple does not mean that it is correct.28,29... 

In carrying out he calculations we use essentially the same procedure as in the case of benzene and naphthalene. As an additional simplification, however, we neglect entirely all the excited states of the molecule, since their contribution to the total energy is comparatively small, and since they would complicate the calculations tremendously if retained. Another slight modification of the procedure is necessitated by the fact that a free radical possesses an odd number of electrons, one of which must remain unpaired. This is taken care of formally by introducing a phantom orbit X with an accompanying phantom electron which is paired with the odd electron.4 In the subsequent... 

The fact that the fuel/air ratio is spatially constant in HCSI engines, at least within a reasonably close approximation, allows substantial simplifications in combustion models. The burn rate or fuel consumption rate dm /dt is expressed as a function of flame surface area the density of the unburnt fuel/air mixture Pu, the laminar burning velocity Sl, and the fluctuations of velocities, i.e., E as a measure of turbulence, u. ... 

PSA. Hence, we only mention here that the definition of PSA is similar to that of Na and N. Thus, PSA possesses the same disadvantages when compared to thermodynamic H-bond descriptors. In fact, it has been estimated that there is a strong linear relationship between the calculated static PSA and the calculated dynamic polar surface [32, 33]. A further possible simplification using only the number of H-bond forming atoms and PLS statistics was proposed [35]. Excellent correlahons of PSA with the number of H-bond donors and acceptors were published [35, 36]. 

In this section, we will investigate the surface structure of the electrode in the potential range before a surface or bulk oxide starts forming, and will restrict ourselves to the adsorption of atomic oxygen only (not OH ) [Jacob and Scheffler, 2007]. Furthermore, in our simulations, we assume a single-crystal Pt(lll) electrode, which will be compared with the experimental CV curve (Fig. 5.9) for poly crystalline Pt. This simplification is motivated by the fact that our interest here is to describe the general behavior of the system only. 

Although Eq. (25) has no physical meaning and although its simplification is not valid thermodynamically, since the condition of electroneutrality in both phases is not fulfilled if the proton cannot partition, it agrees very well with experimental data, which justifies its use. In fact, computerized fitting procedures correct for this error in commercial two-phase titrators, so that the experimental value of p K can directly be introduced in Eq. (25). 

With the second simplification reflecting the fact that insignificant decay of N, will have occurred because t Ti. Because A,i A,2, Equation (4) can be further simplified because... 

The picture of the compact double layer is further complicated by the fact that the assumption that the electrons in the metal are present in a constant concentration which discontinuously decreases to zero at the interface in the direction towards the solution is too gross a simplification. Indeed, Kornyshev, Schmickler, and Vorotyntsev have pointed out that it is necessary to assume that the electron distribution in the metal and its surroundings can be represented by what is called a jellium the positive metal ions represent a fixed layer of positive charges, while the electron plasma spills over the interface into the compact layer, giving rise to a surface dipole. This surface dipole, together with the dipoles of the solvent molecules, produces the total capacity value of the compact double layer. 

On the other hand, neither can we apply some of the simplifications we used in the case of constant detector noise, particularly the fact that the integral of the Normal distribution... 

The chemical behavior of ions, ion pairs, and polarizable molecules partakes of the same indistinctness as the definitions of these species. Any attempt to make a complete catalog of the reactions of ions will almost certainly include borderline reactions whose intermediates are in fact ion-pairs or even covalent molecules. For many purposes the identification of a reaction as carbonium ion-like, or what the Germans would call Krypto-ionenreaktion, is as useful as the certain knowledge that the intermediate is actually a carbonium ion. Many of the ionic reaction mechanisms in the literature do not represent actual free ions and were not so intended by their authors. The ionic representation is often merely a convenient simplification if it is an oversimplification it is one that is easily rectified when the pertinent data become available. The value of such approximate mechanisms is that... 

In spite of all the simplifications Bakker and Van den Akker applied and given the black box approach for the impeller swept domain, their simulations resulted in values for the bubble size just below the liquid surface, overall holdup, and average kfl values which are in good agreement with their experimental data (see Table II). The major step forward they made was the acquisition of the different spatial distributions of average bubble size (see Fig. 13), bubble holdup and kfl as effected by three common impeller types. As a matter of fact, their approach may be restricted to low values of the gas hold-up. 

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