Ionic model assumptions

Here, a model for the retention of small ions in ion exchange chromatography is presented, called the contact layer model. The assumptions behind the model are based on the contact value theorem that is insensitive to both the geometry and to the specific ionic models. The critical assumption is most probably the approximation made when approximating Equation 15.47 by Equation 15.48, an approximation that seems to be satisfied in most experiments. It is evident that when the approximation is not valid, the retention of A will still decrease with an increasing but because of the com-... 

Theoretical aspects of the bond valence model have been discussed by Jansen and Block (1991), Jansen et al. (1992), Burdett and Hawthorne (1993), and Urusov (1995). Recently Preiser et al. (1999) have shown that the rules of the bond valence model can be derived theoretically using the same assumptions as those made for the ionic model. The Coulomb field of an ionic crystal naturally partitions itself into localized chemical bonds whose valence is equal to the flux linking the cation to the anion (Chapter 2). The bond valence model is thus an alternative representation of the ionic model, one based on the electrostatic field rather than energy. The two descriptions are thus equivalent and complementary but, as shown in Section 2.3 and discussed further in Section 14.1.1, both apply equally well to all types of acid-base bonds, covalent as well as ionic. 

This chapter shows that the ionic model can not only be presented in terms of chemical bonds characterized by their electrostatic flux, but also that the improbable assumptions of the model are satisfied by the wide range of compounds that conform to the following two conditions ... 

Why should one go to all this trouble and do all these integrations if there are other, less complex methods available to theorize about ionic solutions The reason is that the correlation function method is open-ended. The equations by which one goes from the gs to properties are not under suspicion. There are no model assumptions in the experimental determination of the g s. This contrasts with the Debye-Htickel theory (limited by the absence of repulsive forces), with Mayer s theory (no misty closure procedures), and even with MD (with its pair potential used as approximations to reality). The correlation function approach can be also used to test any theory in the future because all theories can be made to give g(r) and thereafter, as shown, the properties of ionic solutions. 

The details of the modified electron-gas (MEG) ionic model method have been fully described by Gordon and Kim (1972). The fundamental assumptions of the method are (1) the total electron density at each point is simply the sum of the free-ion densities, with no rearangements or distortion taking place (2) ion-ion interactions are calculated using Coulomb s law, and the free-electron gas approximation is employed to evaluate the electronic kinetic, exchange, and correlation energies (3) the free ions are described by wave functions of Hartree-Fock accuracy. Note that this method does not iterate to a self-consistent electron density. 

Ionic crystals may be viewed quite simply in terms of an electrostatic model of lattices of hard-sphere ions of opposing charges. Although conceptually simple, this model is not completely adequate, and we have seen that modifications must be made in it. First, the bonding is not completely ionic with compounds ranging from the alkali halides, for which complete ionicity is a very good approximation, to compounds for which the assumption of the presence of ions is rather poor. Secondly, the assumption of a perfect, infinite mathematical lattice with no defects is an oversimplification. As with all models, the use of the ionic model does not necessarily imply that it is true , merely that it is convenient and useful, and if proper caution is taken and adjustments are made, it proves to be a fruitful approach. 

The equilibrium constant defined by Eq. (5.24) can be determined directly from experimental c7o(pH) curves obtained at different ionic strengths as their CIP, but Eq. (5.24) is not sufficient to calculate the cro(pH) curves when only the pHo is known, namely, some model assumption is necessary to calculate 0 beyond the pHo. Generally the surface potential makes the changes in uq with pH less steep than they would be (assuming a fixed iVg value) without the exponential term in Eq. (5.24). It was already discussed above (Eq. (5.21)) that the Nernstian 0 leads to constant [=AI0H2 ]/[=A10H ], and consequently constant (tq over the entire pH range, and this is in conflict with experimental facts. 

The fact that radius ratio arguments do not always predict the correct structure is sometimes regarded as a serious failure of the ionic model, and an indication that nonionic forces must be involved in bonding. Given the uncertainties in definition of ionic radii, however, and the fact that they are known to vary with CN, it is hardly surprising that predictions based on the assumption of hard spheres are unreliable. It also appears that for some compounds the difference in energy between different structure types is very small, and the observed structure may change with temperature or pressure. 

Certain model assumptions are necessary in order to reveal the surface concentration of specifically adsorbed ions in the total surface excess F,-. Usually, the ionic component of the electrical double layer (EDL) is assumed to consist of the dense part and the diffuse layer separated by the so-called outer Helmholtz plane. Only specifically adsorbing ions can penetrate into the dense layer close to the surface (e.g. iodide ions), with their electric centers located on the inner Helmholtz plane. The charge density of these specifically adsorbed ions ai is determined by their surface concentration F Namely, for single-charged anions ... 

It is important to recognize the distinction between a theoretical value and a literature value discrepancies between an experimentally determined value and a literature value are due to random and systematic errors, but a discrepancy between an experimental value and a theoretical value may be indicative of inappropriate assumptions in the theoretical model. For example, calculated lattice enthalpies usually assume a purely ionic model and therefore a discrepancy with experiment may indicate a covalent character in the bonding. 

This derivation of the ionic model provides not only a natural definition of a bond, but it also defines the scope of the model. The assumptions on which the derivation is based show that far from being confined to compotmds whose bonds have ionic character, the ionic model can be used for any valence compound. It can be used to describe not just NaCl but also SFs, CO2, CH4, CH3COOH, and O2, all of which have networks with bipartite graphs. 

In bond valence theory, atoms (with one exception) are always treated as uncharged. A bond is formed when the valence shells of two neutral atoms overlap. The electrons from the two atoms spin-pair, but they are still counted as being part of their original spherically symmetric valence shells. This assumption simplifies the description of the bond since it avoids using such elusive concepts as electron transfer, ionic character, and atomic charge. The only exception to this assumption is the ionic model which is derived from the bond valence theory in Sect. 5. Even in this case the transfer of electrons from the cation to the anion is a matter of formal bookkeeping it does not imply any physical movement of the electrons. 

The correlation functions of the partly quenched system satisfy a set of replica Ornstein-Zernike equations (21)-(23). Each of them is a 2 x 2 matrix equation for the model in question. As in previous studies of ionic systems (see, e.g.. Refs. 69, 70), we denote the long-range terms of the pair correlation functions in ROZ equations by qij. Here we apply a linearized theory and assume that the long-range terms of the direct correlation functions are equal to the Coulomb potentials which are given by Eqs. (53)-(55). This assumption represents the mean spherical approximation for the model in question. Most importantly, (r) = 0 as mentioned before, the particles from different replicas do not interact. However, q]f r) 7 0 these functions describe screening effects of the ion-ion interactions between ions from different replicas mediated by the presence of charged obstacles, i.e., via the matrix. The functions q j (r) need to be obtained to apply them for proper renormalization of the ROZ equations for systems made of nonpoint ions. 

The type of disorder may be determined by conductivity measurements of electronic and ionic defects as a function of the activity of the neutral mobile component [3]. The data are commonly plotted as Brouwer diagrams of the logarithm of the concentration of all species as a function of the logarithm of the activity of the neutral mobile component. The slope is fitted to the assumption of a specific defect-type model. 

In polymer electrolytes (even prevailingly crystalline), most of ions are transported via the mobile amorphous regions. The ion conduction should therefore be related to viscoelastic properties of the polymeric host and described by models analogous to that for ion transport in liquids. These include either the free volume model or the configurational entropy model . The former is based on the assumption that thermal fluctuations of the polymer skeleton open occasionally free volumes into which the ionic (or other) species can migrate. For classical liquid electrolytes, the free volume per molecule, vf, is defined as ... 

All the above derivations are based on the assumption of a single ionic species moving through the oxide. The implications of such an approach have been considered most thoroughly by Dignam.47 The present state of the art in the field of ionic conduction modeling needs improvement. The theory should include the following ... 

The growth of an anodic alumina film, at a constant current, is characterized by a virtually linear increase of the electrode potential with time, exemplified by Fig. 10, with a more or less notable curvature (or an intercept of the extrapolated straight line) at the beginning of anodization.73 This reflects the constant rate of increase of the film thickness. Indeed, a linear relationship was found experimentally between the potential and the inverse capacitance78 (the latter reflecting the thickness in a model of a parallel-plate capacitor under the assumption of a constant dielectric permittivity). This is foreseen by applying Eq. (38) to Eq. (35). It is a consequence of the need for a constant electric field on the film in order to transport constant ionic current, as required by Eqs. (39)-(43). 

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