Phenomenological Treatment

What is an ion-pair In order to shed light on the theory that governs ion-pairing, the process first will be treated phenomenologically. 

The effects of ion-pairing have been noted in countless experimental situations involving conductometry, potentiometry, spectroscopy, solvent extraction, separative techniques, activity measurement, and kinetic behavior among others. As far as chromatography is concerned, the electrical neutrality and the increased lipo-philicity of ion-pairs, compared to unpaired ions, are features of utmost importance involved in retention adjustment. 

The physics of electrostatics in aqueous solution has attracted scientists notice for centuries. At present, the solvation of ions, volumes and radii of ions in solution, and ionic interactions are still hotly debated research fields. Recently, thermodynamic, transport and structural data were mutually employed for gaining fruitful physicochemical insights into electrolyte solutions [2]. This chapter recapitulates the essential 

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Models of a second type (Sec. IV) restrict themselves to a few very basic ingredients, e.g., the repulsion between oil and water and the orientation of the amphiphiles. They are less versatile than chain models and have to be specified in view of the particular problem one has in mind. On the other hand, they allow an efficient study of structures on intermediate length and time scales, while still establishing a connection with microscopic properties of the materials. Hence, they bridge between the microscopic approaches and the more phenomenological treatments which will be described below. Various microscopic models of this type have been constructed and used to study phase transitions in the bulk of amphiphihc systems, internal phase transitions in monolayers and bilayers, interfacial properties, and dynamical aspects such as the kinetics of phase separation between water and oil in the presence of amphiphiles. 

Further references to the phenomenological treatment may be found in Treloar [16], Ogden [35], Erman and Mark [5], and Mark [47],... 

The phenomenological treatment assumes that the Gibbs energies of activation Gox and Gred depend on the electrode potential , but that the pre-exponential factor A does not. We expand the energy of activation about the standard equilibrium potential >0o of the redox reaction keeping terms up to first order, we obtain for the anodic reaction ... 

Traditionally, electron transfer reactions have been treated using chemical kinetics concepts. We briefly review the phenomenological treatment to introduce some concepts that will be useful later. 

This phenomenological treatment, however, can be extended to include other quenching mechanisms. For example, living polymers are known to quench when the monomers are reacted to completion. In the context of the iGLE, a friction kernel that would simulate such a mechanism is the addition of the term. 

The phenomenological treatment allows us to identify the diffusion coefficients with the gradients in chemical potentials such that 3)... 

Figure 5-11 illustrates the results of an oxide interdiffusion experiment. Clearly, the transport coefficients are not single valued functions of composition. From the data, one concludes that for a given composition, the chemical diffusion coefficients depend both on time and location in the sample [G. Kutsche, H. Schmalzried (1990)]. Let us analyze this interdiffusion process in the ternary solid solution Co. O-Nq. O, which contains all the elements necessary for a phenomenological treatment of chemical transport in crystals. The large oxygen ions are almost immobile and so interdiffusion occurs only in the cation sublattice of the fee crystal. When we consider the following set ( ) of structure elements... 

When we deal with the kinetics of hydride reactions we have to be aware that hydride thermodynamics cannot be properly formulated without taking into account the (relative) immobility of the metal component. This immobility can sometimes render the interpretation of the experimental reaction kinetics ambiguous. With this difficulty in mind, let us outline concepts which describe the kinetics of hydride formation and decomposition. An extensive account, including a first order phenomenological treatment, has been given by [P.S. Rudman (1983)]. The conceptual framework for a more rigorous discussion is found in, for example, [G. B. Stephenson (1988)]. 

The present article attempts to clarify the nature of the discontinuous transition of gels. First, in Sect. 2 we give an outline of the fundamental aspect of the volume phase transition on the basis of the Flory-Rehner theory of gels, with special attention to how the discontinuous transition comes about within the phenomenological treatment Then, in Sect 3 previous experimental results... 

Our quantitative approach follows the phenomenological treatment usually given for conformational isomers. Consider the free-energy... 

This phenomenological treatment first employed in the mean spherical approximation [Eq.(43)J, in a form suitable for a potential made up of soft core and an attractive part [49, 50], has proven its reliability [22]. 

A phenomenological treatment of the hydration repulsion, based on a Landau expansion of the free energy density, was proposed by Marcelya and Radic.9 They showed that, if the free energy density is a function of an order parameter that varies continuously from the surface, and if only the quadratic terms in this parameter and its derivative are nonnegligible, an exponential decay... 

A simplified version of the theory was presented by Cevc and Marsh.12 They started from the Marcelja—Radic phenomenological treatment, assumed that the polarization constitutes the order parameter, and used the Gruen—Marcelja model to explain the various contributions to the free energy density. 

While actual chemical events involved in nucleation and crystal growth are not known a phenomenological treatment (gives some insight. Willard Gibbs (9J considered processes of phase separation of two extreme kinds. In the first, fluctuations in concentration occur which are minute in volume but large in extent of departure from the mean (the case of binodal phase separation). In the second the volume of the fluctuation is large but the deviation from the mean for the solution is minute (responsible for spinodal phase separation). In nucleation of zeolites one is conerned only with fluctuations of the first kind. 

A phenomenological treatment of this nucleation process was developed several years ago by N. Cabrera and M. Levine (2, 3). According to this treatment the activation energy for nucleation at a dislocation AF is smaller than that on a perfect surface because of the extra energy localized around the dislocation. This difference is only significant at high under satura-tions, at which AF decreases and becomes ultimately zero. 

How many ions travel a distance a , how many, x, etc. In other words, how are the ions spatially distributed after a time t, and how does the spatial distribution vary with time This spatial distribution of ions will be analyzed, but only after a phenomenological treatment of o steady-state diffusion is presented. 

This is the Einstein relation. It is probably the most important relation in the theory of the movements and drift of ions, atoms, molecules, and other submicroscopic particles. It has been daived ho e in an atomistic way. It will be recalled that in the phenomenological treatment of the diffusion coefficient (Section 4.2.3), it was shown that... 

In the phenomenological treatment of the directed drift that the field brings, we take the attitude that there is a stream of cations going toward the negative electrode and anions going toward the positive one. We now neglect the random diffusive movements they do not contribute to the vectorial flow that produces an electrical current. 

In the phenomenological treatment of conduction (Section 4.2.12), it was stated that the equivalent conductivity A varies with the concentration c of the electrolyte according to the empirical law of Kohlrausch Piq. (4.139)]... 

When the primary acceptor is prereduced, electron spin polarization can transfer by exchange interaction from BPh to Q, leading to an inversion of the EPR line of in RCs where was magnetically uncoupled from Fe [112] (Fig. 8). From a phenomenological treatment [112-114] it was concluded that the exchange interaction /(BPh QA) was 3 - 5 G, whereas /(P BPh ) was between 1 and 5 G. A more sophisticated treatment of the three-spin system P BPh QX [115] led to 7(P BPh ) between 0 and +8 G. (Note that for / = 0 polarization may develop if D = 0.) A positive value of / for a biradical state is unusual it might be explained by some form of superexchange via an intermediate (possibly one of the accessory bacteriochlorophylls). 

Phenomenological treatments which approximate the molecular potential field (Born-Oppenheimer approximation) by a series of classical energy equations and adjustable parameters. These treatments may be called classical mechanical only in the sense that harmonic force-field expressions stemming from vibrational analysis methods are often introduced, though strictly speaking one is free to select any set of functions that reproduces the experimental data whitin chosen limits of accuracy. 

In Chapters I, X, and XI it is stressed that the microscopic derivation of equations such as some of those used here should be discussed carefully. This is to avoid some ambiguous features of a purely phenomenological treatment. However, as these are widely used in the literature of stochastic processes, we shall show how to approach the problem of their solution while avoiding those difficulties by using a more rigorously founded microscopic derivation (see Chapters X and XI). 

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