Quantitative Treatments

The one electrMi transfer processes were quantitatively discussed for the regime of the square-wave pulsating overpotential [31, 32], The square-wave pulsating overpotential is described by Eqs. (4.9), (4.10), and (4.11), as weU as by Eq. (4.51) [18, 31]  

Assuming that at sufficiently high frequencies the surface concentration in the pulsating overpotential deposition does not vaiy with time, it is easy to show that response of the current density, i, to the input overpotential  

The right-hand side of Eq. (4.51) should be transformed by taking Eq. (4.53) into account. The output current during pauses (t/ = 0) becomes  

It is easy to show that the difference between the current density on the flat surface and at the tip of the dendrites during the off period is given by  

The approximation methods which throughout find application in wave-mechanical calculations are the perturbation method and the variational method. In the former, one starts from a simple case for which the solution (unperturbed state) is known, in our case the hydrogen atom. Then the actual situation is regarded as a consequence of a (small) perturbation of the first-mentioned state, in our case the perturbation by the field of the second proton. 

The first method, developed for the calculation of planetary orbits, has also found much application in the wave mechanics of the atom. The second method is sometimes physically less clear but in most cases this method leads with less calculation more rapidly and more accurately to the end in view than the former method. In the wave-mechanical treatment of molecules the variational method is indeed mainly used. 

In most cases one chooses as the variational function a linear combination of wave functions which are correct to a certain approximation, for example, for large nuclear separations. In this case we choose (see below) 0 = cx(pL + r29n. 

Such values must now be sought for the coefficients cx and c2 that the (apparent) energy E corresponding to the variational function 0 is as low as possible. We find the conditions for this by differentiating E with respect to cx or c2, since we have for the minimum value of E with respect to cx or c2  

The above-mentioned quantity / 0 H 0 dv must still be divided by / 02 dv since 0 is not normalized but 9 is. The expression for the energy becomes (p. 123)  

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A quantitative treatment for the depletive adsorption of iogenic species on semiconductors is that known as the boundary layer theory [84,184], in which it is assumed that, as a result of adsorption, a charged layer is formed. Doublelayer theory is applied, and it turns out that the change in surface potential due to adsorption of such a species is proportional to the square of the amount adsorbed. The important point is that very little adsorption, e.g., a 0 of about 0.003, can produce a volt or more potential change. See Ref. 185 for a review. 

Abel E W, Coston T P J, Orrell K G, Sik V and Stephenson D 1986 Two-dimensional NMR exohange speotrosoopy. Quantitative treatment of multisite exohanging systems J. Magn. Reson. 70 34-53... 

A quantitative treatment of surfactant solubility has been successfully made empirically using linear free energy relationships. An important relation is that for the linear free energy of transfer of alkanes to water [23] ... 

It should be noted that none of the foregoing equations relates to stoichiometric concentrations of additives. Quantitative treatment is precluded by ignorance of the effects of ionic atmosphere and of ionpairing in these media. 

Before concluding this section, there is one additional thermodynamic factor to be mentioned which also has the effect of lowering. Since we shall not describe the thermodynamics of polymer solutions until Chap. 8, a quantitative treatment is inappropriate at this point. However, some relationships familiar from the behavior of low molecular weight compounds may be borrowed for qualitative discussion. The specific effect we consider is that of chain ends. The position we take is that they are foreign species from the viewpoint of crystallization. 

The molecular weight distribution for a polymer like that described above is remarkably narrow compared to free-radical polymerization or even to ionic polymerization in which transfer or termination occurs. The sharpness arises from the nearly simultaneous initiation of all chains and the fact that all active centers grow as long as monomer is present. The following steps outline a quantitative treatment of this effect ... 

The quantitative treatment of solubiUty is based on the familiar free energy equation governing mutual miscibility ... 

In the presence of 6-iodo-l-phenyl-l-hexyne, the current increases in the cathodic (negative potential going) direction because the hexyne catalyticaHy regenerates the nickel(II) complex. The absence of the nickel(I) complex precludes an anodic wave upon reversal of the sweep direction there is nothing to reduce. If the catalytic process were slow enough it would be possible to recover the anodic wave by increasing the sweep rate to a value so fast that the reduced species (the nickel(I) complex) would be reoxidized before it could react with the hexyne. A quantitative treatment of the data, collected at several sweep rates, could then be used to calculate the rate constant for the catalytic reaction at the electrode surface. Such rate constants may be substantially different from those measured in the bulk of the solution. The chemical and electrochemical reactions involved are... 

Dispersion Characteristics The chief characteristics of gas-in-liquid dispersions, like those of hquid-in-gas suspensions, are heterogeneity and instabihty. The composition and structure of an unstable dispersion must be obsei ved in the dynamic situation by looking at the mixture, with or without the aid of optical devices, or by photographing it, preferably in nominal steady state photographs usually are required for quantitative treatment. Stable foams may be examined after the fact of their creation if they are sufficiently robust or if an immobilizing technique such as freezing is employed [Chang et al., Ind. Eng Chem., 48, 2035 (1956)]. 

Several methods of quantitative description of molecular structure based on the concepts of valence bond theory have been developed. These methods employ orbitals similar to localized valence bond orbitals, but permitting modest delocalization. These orbitals allow many fewer structures to be considered and remove the need for incorporating many ionic structures, in agreement with chemical intuition. To date, these methods have not been as widely applied in organic chemistry as MO calculations. They have, however, been successfully applied to fundamental structural issues. For example, successful quantitative treatments of the structure and energy of benzene and its heterocyclic analogs have been developed. It remains to be seen whether computations based on DFT and modem valence bond theory will come to rival the widely used MO programs in analysis and interpretation of stmcture and reactivity. 

Quantitative Treatment of Resistance to Mass Transfer Dispersion... 

At a deeper level, the reaction mechanism requires a quantitative treatment of... 

In the original quantitative treatment of the pH dependence of the kinetics, it was necessary to make the assumption that k K2 From Eq. (3-187), this... 

At some stage between cases 2 and 3, coalescence into a single broadened band takes place. A full quantitative treatment requires nonlinear regression of the line shape to the theoretical relationship. 

The Henderson-Hasselbalch equation provides a general solution to the quantitative treatment of acid-base equilibria in biological systems. Table 2.4 gives the acid dissociation constants and values for some weak electrolytes of biochemical interest. 

Before beginning a quantitative treatment of enzyme kinetics, it will be fruitful to review briefly some basic principles of chemical kinetics. Chemical kinetics is the study of the rates of chemical reactions. Consider a reaction of overall stoichiometry... 

The problems involved in attempts to develop quantitative treatments of organic chemistry are discussed. An improved version (MINDO/3) of the MINDO semiempirical SCF-MO treatment is described. Results obtained for a large number of molecules are summarized. 

Reactions involving monocyclic six-membered heteroaromatic rings have not been studied sufficiently extensively to allow a quantitative treatment of substituent effects. However, comparison with aza-naphthalene reactivities indicates that aza- and polyaza-benzene systems must also be highly selective. 

The kinetic mechanism of emulsion polymerization was developed by Smith and Ewart [10]. The quantitative treatment of this mechanism was made by using Har-kin s Micellar Theory [18,19]. By means of quantitative treatment, the researchers obtained an expression in which the particle number was expressed as a function of emulsifier concentration, initiation, and polymerization rates. This expression was derived for the systems including the monomers with low water solubility and partly solubilized within the micelles formed by emulsifiers having low critical micelle concentration (CMC) values [10]. 

In this sub-section it is intended first to outline the theoretical basis of these diagrams by considering a simple metal-/4-gas-5 binary system followed by a quantitative treatment of a hypothetical metal A/(at. wt. 50) and oxygen binary system. Finally the application of these diagrams will be... 

A1C13, or S02 in an inert solvent cause colour changes in indicators similar to those produced by hydrochloric acid, and these changes are reversed by bases so that titrations can be carried out. Compounds of the type of BF3 are usually described as Lewis acids or electron acceptors. The Lewis bases (e.g. ammonia, pyridine) are virtually identical with the Bransted-Lowry bases. The great disadvantage of the Lewis definition of acids is that, unlike proton-transfer reactions, it is incapable of general quantitative treatment. 

For further development of the quantitative treatment of the desorption kinetics, the work of Redhead (31) and of Carter (32) is of great impor-... 

A quantitative treatment based on the following approach has been recently given to the idea of explaining the multiplicity of desorption spectra by the existence of different desorption mechanisms rather than by different adsorption states (98, 117). Consider a surface on which an adsorption equilibrium has been established at a given temperature. On heating the surface, desorption occurs, the probability of which is composed of at... 

From the various possible geometric shapes of reactant crystallites, discussion here will be restricted to a consideration of reaction proceeding in rectangular plates knd in spheres [28]. A complication in the quantitative treatment of such rate processes is that reaction in those crystallites which were nucleated first may be completed before other particles have been nucleated. Due allowance for this termination of interface advance, resulting from the finite size of reactant fragments accompanied by slow nucleation, is incorporated into the geometric analysis below. 

A similar quantitative treatment of sulphoxides as hydrogen bonding acceptors has been obtained by comparing the IR frequency shift AvOH of the C—I bond in an acetylenic iodide such as IC=CI (Avc j) due to formation of a C—T complex with phenol in various bases. This investigation suggests that sulphoxides belong to the same family as carbonyls, phosphine oxides, arsine oxides and their derivatives90. 

What Do We Need to Know Already This chapter extends the thermodynamic discussion presented in Chapter 7. In particular, it builds on the concept of Gibbs free energy (Section 7.12), its relation to maximum nonexpansion work (Section 7.14), and the dependence of the reaction Gibbs free energy on the reaction quotient (Section 9.3). For a review of redox reactions, see Section K. To prepare for the quantitative treatment of electrolysis, review stoichiometry in Section L. 

The activated complex can be described as involving resonance of the fourth bond of carbon between the hydroxyl and iodine ions. Some very interesting rough quantum-mechanical calculations bearing on the theory of chemical reactions have been made of Eyring and Polanyi and their collaborators. It is to be hoped that the quantitative treatments can be made more precise and more-reliable but before this can be done effectively there must take place the extensive development of the qualitative theory of chemical reactions, probably in terms of resonance. 

A qualitative explanation of these abnormally large diamagnetic susceptibilities as arising from the Larmor precession of electrons in orbits including many nuclei3 has come to be generally accepted. With the aid of simple assumptions, I have now developed this idea into an approximate quantitative treatment, described below. 

When one compares the brutto polymerization rate constants, a measure of the reactivity of monomers during cationic homopolymerizations is obtained. It was found for p-substituted styrenes that lg kBr increased parallel to the reactivity, which the monomers show versus a constant acceptor 93). The reactivity graduation of the cationic chain ends is apparently overcomed by the structural influence on the monomers during the entire process of the cationic polymerization. The quantitative treatment of the substituent influences with the assistance of the LFE principle leads to the following Hammett-type equations for the brutto polymerization rate constants ... 

A general theory based on the quantitative treatment of the reaction layer profile exists for pure redox catalysis where the crucial function of the redox mediator is solely electron transfer and where the catalytic activity largely depends only on the redox potential and not on the structure of the catalyst This theory is consistent... 

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