Common Simplified Theoretical Study

Commonly, the study of a redox titration curve is achieved as follows. The first step is to recognize the quantitative character of the titration reaction, which must be 

Before the equivalence point, Y X. The present species are the remaining ions Fe + and the Fe + and Ce formed. Applying relations (17.5)-(17.7) and assuming activities equal to the corresponding concentrations give 

As a result, if we accept the hypothesis that the titration reaction is complete, we see that the equilibrium potential value and the shape of the curve depend only on the couple Fe +/Fe + through its standard potential value E° (Fe +/Fe +). Some authors say that the potential value is imposed by the couple Fe +/Fe +. The first point (cp = 0) is a particular one. The only species existing in the titration vessel is Fe +. The potential value is ill defined and is of no practical consequence. 

After the equivalence point (Y X q) 1), the present species are Fe, Ce , and Ce (which is in excess). Fe + no longer exists since the titration reaction may be considered complete. After handling Eqs. (17.4)-(17.7), we find 

At the equivalence point, we can think, a priori, that the potential Eep must depend on both couples, that is, on the values ° (Fe +/Fe +) and ° (Ce /Ce ), since just before it, the potential value is imposed only by one couple and just after it by the other one. The equivalence point is defined exactly by the relations X = Y and tp = 1. Due to the quantitative character of the titration reaction, it is clear that only a very weak concentration [eFe ] of ferrous ions has not reacted. Thus, according to Eqs. (17.5)-(17.7), there exists an equal concentration [eCe ] of Ce that has not reacted at the equivalence point  

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The modem literature contains a number of theoretical studies, where attempts have been made to describe the combined effect of reaction and viscous micro mixing in mathematical models. From experimental studies both the reaction and the mixing parameters could then be determined. Each of these theories is based on a simplified physical picture of the flow and mixing phenomena taking place in the reactor. The published models have a few things in common ... 

There are several types of interfaces that are of great practical importance and that will be discussed in turn. These general classifications include, solid-vacuum, liquid-vacuum, solid-gas, liquid-gas, solid-liquid, liquid-liquid, and solid-solid. From a practical standpoint, solid- and liquid-vacuum interfaces are of little concern. They are most often encountered in the context of theoretical derivations, since the absence of a second phase simplifies matters greatly, or in studies of high-vacuum processes such as deposition, and sputtering. The true two-phase systems (assuming that a vacuum is not considered to be a true phase ) are the ones which are of most importance in practical applications and that are addressed in most detail here. A list of commonly encountered examples of these interfaces is given in Table 2.1. 

As mentioned earlier, most studies of field interactions with liquid crystals are done using thin films with a well-defined initial state, usually a monodomain or a thin film with a simple distortion induced by incommensurate surface anchoring. These conditions simplify observation and theoretical analysis. However, most liquid crystal materials that are not specially prepared contain topological defects that are very important to their response to external fields. One class of defect commonly observed in nematics is the disclinalion line. At a disclination line the director field is ill defined. The director field turns around the disclination line a multiple of half-integer times. Several disclination lines are shown in Fig. 8. 

Even though some progress has been made towards understanding electrocatalytic process and screening electrocatalysts from DFT, the method has difficulty in providing quantitative numbers for detailed reaction steps. On one hand, methodological improvements are required to describe the electron transfer at solid-liquid interface, the band structure, and the excited states effectively, which is currently limitation of DFT. On another hand, the model systems in DFT studies are somewhat too simplified to model the real catalysts effectively. For instance, the real catalysts are powders, which may behave differently with size. Recently, efforts have been made to model the nanoparticles with the size of the real catalysts (<5 nm), showing indeed different behaviors from the extended surfaces even in term of trend (Fig. 3), a common model used in DFT studies [24, 25]. Thus, theoretical... 

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