The polarizability of ions

The effect of polarization is evident in the gaseous alkali-halide molecules, being most pronounced for small and large X (Lil) and large M and small X (CsF), and resulting in a dipole moment appreciably less than the product of the internuclear distance d) and the electronic charge (e)  

We shall discuss here some general features of the structures of crystalline halides AX, AX2, and AX3,the structures of which are described in more detail in Chapter 9. 

Dihalides and trihalides. Here we comment only on those features of the structures which indicate the importance of polarization effects and the inadequacy of the simple hard sphere treatment for halides other than fluorides. 

Type of structure AX2 Octahedra sharing Coordination ofX Octahedra sharing Type of structure AX3 

3D rutile CaCl2 ) 2 edges and j 6 vertices Triangular Pyramidal Linear Non-linear 6 vertices 6 vertices 3D Re03 RhF3 

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This method may also be applied to the calculation of polarizabilities. Indeed, we arrived at the present formulation of this method when wishing to perform such calculations. More recently,7 it has been applied to the problem of calculating the polarizability of ions having the rare gas structure. Here again the ground state is considered as being described by an antisymmetric function composed of Slater orbitals. The excited orbital was taken to be of the form ... 

Variables 2 and Zj are charges of ions i and j Ay is the Pauling factor defined as Ay = (1 + zjnx + z-Jn i, where nK and nj represent the numbers of electrons in the outermost shell of ions i and j, respectively Cy = (3/2) aiajEiEj/(Ei + Ej) and dy = (9/4e2)Cy(a, 1/Ar1 + atjE/Nj), where a denotes the polarizability of ions, N is the number of the total electrons of an ion, and E is the first ionization potential, evaluated from the Equation Ef = Nle2h2I Tr2mai for ion i, where h and m are the Planck constant and the mass of the ion, respectively. Values of p, b, and cr are estimated from isothermal compressibilities and thermal expansion coefficients of 17 rock-salt-type crystals of alkali halides by Fumi and Tosi (15). 

The polarizability of ions forming the melt gives rise to light scattering either due to electrostatic field fluctuation around the ion and/or to polarizability changes due to bond formation within the melt (i.e. complex formation). 

As noted above, Kossel introduced the idea that the transition from ionic to covalent substances is gradual, the covalence increasing with the mutual polarizing influence of ions. This idea was developed by Fajans and his school who defined the polarizabilities of ions and estimated the polarizing action of cations (Z/r ), but ultimately failed to create a quantitative theory. The reason is obvious [10] there are no completely ionic substances, only intermediate cases, more or less approaching this type. Hence the parameters of ideal ions are not available experimentally, the more so since ionic radii cannot be uniquely defined from interatomic distances (see Chap. 1). Thus the polarization concept remained only qualitative. However, the contribution in the bond energy of the polarizing effect of atoms can be described in the form that has proven itself for the van der Waals interaction (see Sect. 4.4), where the deviation of the A B distance from the mean of A A and B B distances is a function of the difference of the atomic polarizabilities... 

The polarizability of ions and molecules was in fact introduced by Fajans32 as early as in 1923, but it is understood that various other factors contribute to hard and soft properties. 

Atoms exist in a molecule in the form of ions no matter whether they arc associated in ionic or co-valent bonds. Like the electrons moving under an electric field, ions can move along the direction of the electric field too. Ions of opposite charges are bound together through chemical bonds, forming a molecule. Under an electric field, those ions will move to the opposite direction, creating dipole moment. The polarizability of ions, a , can be expressed as [5] ... 

This work shows that the use of the RIM allows to describe the local structure of pure silica and a silica-calcia mixture for which measurements are complicated or not possible. However, it has been shown that the calculation of properties such as density exhibit some differences from measurements but the trend is correct and offers, in a first approach, a rather good estimation of it at elevated temperature. For complex systems, including multi-cations with different oxidation degrees, it appears that the polarizability of ions should be included to get more reliable results. 

The solubilities of the ionic halides are determined by a variety of factors, especially the lattice enthalpy and enthalpy of hydration. There is a delicate balance between the two factors, with the lattice enthalpy usually being the determining one. Lattice enthalpies decrease from chloride to iodide, so water molecules can more readily separate the ions in the latter. Less ionic halides, such as the silver halides, generally have a much lower solubility, and the trend in solubility is the reverse of the more ionic halides. For the less ionic halides, the covalent character of the bond allows the ion pairs to persist in water. The ions are not easily hydrated, making them less soluble. The polarizability of the halide ions and the covalency of their bonding increases down the group. 

According to theory (22), the rate constant for reaction between an ion and a molecule can be expressed in terms of the polarizability of the neutral species and of the reduced mass of the reacting pair... 

Thus, in the systems under consideration, MeX may form haionium ions with growing carbenium ions. Since the stability of haionium ions depends on the polarizability of ttie halogen38 —I > —Br > —Cl, Mel should form the most stable haionium ions, le., have most pronounced poisoning effect, followed by MeBr and MeCl. Indeed, Mel may even compete for the carbocation with highly nucleophilic counterions. 

Considerable effort has been expended on Ag atoms and small, silver clusters. Bates and Gruen (10) studied the spectra of sputtered silver atoms (a metal target was bombarded with a beam of 2-keV, argon ions produced with a sputter ion-gun) isolated in D, Ne, and N2. They found that an inverse relationship between Zett of the metal atom and the polarizability of rare-gas matrices (as determined from examination of... 

Here, the last term accounts for the excess ions in the interfacial region, which compensate the excess charge on the electrode surface and keep the overall interface electroneutral. What in electrochemical terms is often described as a polarizable active electrode and an unpolarizable reference electrode ensures that any change of the number of ions in the electrochemical half-cell under consideration, caused by an electrochemical reaction, is just compensated by a corresponding counter-reaction at the reference electrode. 

Heterogeneous ET reactions at polarizable liquid-liquid interfaces have been mainly approached from current potential relationships. In this respect, a rather important issue is to minimize the contribution of ion-transfer reactions to the current responses associated with the ET step. This requirement has been recognized by several authors [43,62,67-72]. Firstly, reactants and products should remain in their respective phases within the potential range where the ET process takes place. In addition to redox stability, the supporting electrolytes should also provide an appropriate potential window for the redox reaction. According to Eqs. (2) and (3), the redox potentials of the species involved in the ET should match in a way that the formal electron-transfer potential occurs within the potential window established by the transfer of the ionic species present at the liquid-liquid junction. The results shown in Figs. 1 and 2 provide an example of voltammetric ET responses when the above conditions are fulfilled. A difference of approximately 150 mV is observed between Ao et A" (.+. ... 

A multitude of semiempirical and semiclassical theories have been developed to calculate electron impact ionization cross sections of atoms and atomic ions, with relatively few for the more complicated case of molecular electron impact ionization cross sections. One of the earlier treatments of molecular targets was that of Jain and Khare.38 Two of the more successful recent approaches are the method proposed by Deutsch and Mark and coworkers12-14 and the binary-encounter Bethe method developed by Kim and Rudd.15,16 The observation of a strong correlation between the maximum in the ionization efficiency curve and the polarizability of the target resulted in the semiempirical polarizability model which depends only on the polarizability, ionization potential, and maximum electron impact ionization cross section of the target molecule.39,40 These and other methods will be considered in detail below. 

The electric field or ionic term corresponds to an ideal parallel-plate capacitor, with potential drop g (ion) = qMd/4ire. Itincludes a contribution from the polarizability of the electrolyte, since the dielectric constant is included in the expression. The distance d between the layers of charge is often taken to be from the outer Helmholtz plane (distance of closest approach of ions in solution to the metal in the absence of specific adsorption) to the position of the image charge in the metal a model for the metal is required to define this position properly. The capacitance per unit area of the ideal capacitor is a constant, e/Aird, often written as Klon. The contribution to 1/C is 1 /Klon this term is much less important in the sum (larger capacitance) than the other two contributions.2... 

The background dielectric constant e for the metal arises from the polarizability of the ion cores and the contribution of interband transitions.11 For mercury and other simple metals, with a large band gap and relatively unpolarizable ion cores, one expects a background dielectric constant close to unity. With e = 1 and n°° = 8.17 x 1022 cm-3 (mercury), the capacitance per unit area is... 

A. R. Ruffa, Theory of the Electronic Polarizabilities of Ions in Crystals Application to the Alkali Halide Crystals, Phys. Rev., 130,1412 (1963). 

Although we have illustrated the effects of polarizability of ions by considering a few cases where the effects are large, there must be some polarization effect for any combination of ions. However, there is an even more important consideration. It is known that the apparent radius of a given ion depends somewhat on the environment of the ion. For example, an ion surrounded by four nearest neighbors will appear to be slightly different in size from one that is surrounded by six ions of opposite charge. We have treated the ionic radius as if it were a fixed number that is the same in any type of... 

As was discussed in Chapter 6, the electronic polarizability, a, of species is very useful for correlating many chemical and physical properties. Values of a are usually expressed in cm3 per unit (atom, ion, or molecule). Because atomic dimensions are conveniently expressed in angstroms, the polarizability is also expressed as A3, so lCT24cm3 = 1 A3. The polarizability gives a measure of the ability of the electron cloud of a species to be distorted so it is also related to the hard-soft character of the species in a qualitative way. Table 9.6 gives the polarizabilities for ions and molecules. 

It is also interesting to note that metal ions having low polarizability (Al3+ Be2+ etc.) are those that are acidic (as shown in Eq. (9.17)). Also, in Chapter 7 we discussed how the polarization of ions leads to a lattice energy that is higher than that predicted on the basis of electrostatic interactions alone. The polarizability data shown in the table make it easy to see that certain ions are much more polarizable than others. Although we will not visit again all of the ramifications of electronic polarizability, it is a very useful and important property of molecules and ions that relates to both chemical and physical behavior. 

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