Ionic radius, definition

Equation 6.1 is valid for a macroscopic particle moving in a continuous medium. In electrophoresis where the analyte ion moves in the media where particle size is comparable with that of the analyte size, this is definitely not the case. Also, analyte ions are not spherical and the term of the ionic radius, the value of which is difficult to estimate, becomes ambiguous. Thus, even in... 

Since an a priori definition of the effective region is hardly possible, each atomic region is usually approximated by a spherical region around the atom, where the radius is taken as its ionic, atomic, or covalent bond radius. The radial distribution of electron density around an atom is also useful to estimate the effective radius of an atom, particularly in ionic crystals. In an ionic crystal, the distance from the metal nucleus to the minimum in the radial distribution curve generally corresponds to the ionic radius. As an example, the radial distribution curves around K in o-KvCrO., (85) are shown in Fig. 19a. The radial distributions of valence electrons (2p electrons) exhibit a minimum at 1.60 A for K(l) and 1.52 A for K(2), respectively. These distances correspond to the ionic radii in crystals (1.52-1.65 A)... 

The definition of crystal radii from the location of the minimum of the experimental electron density between neighbouring ions appears to be physically satisfactory when the individual ions approximate to spherical shape and show little overlap, as is the case in sodium chloride. Where deviations from spherical symmetry become more significant and the zone of electron cloud overlap is appreciable, the concept of ionic radius becomes dubious. 

Partial molar entropies of ions can, for example, be calculated assuming S (H+) = 0. Alternatively, because K+ and Cl ions are isoelectronic and have similar radii, the ionic properties of these ions in solution can be equated, e.g. analysis of B-viscosity coefficients (Gurney, 1953). In other cases, a particular theoretical treatment which relates solvation parameters to ionic radii indicates how the subdivision could be made. For example, the Bom equation requires that AGf (ion) be proportional to the reciprocal of the ionic radius (Friedman and Krishnan, 1973b). However, this approach involves new problems associated with the definition of ionic radius (Stem and Amis, 1959). In another approach to this problem, the properties of a series of salts in solution are plotted in such a way that the value for a common ion is obtained as the intercept. For example, when the partial molar volumes of some alkylammonium iodides, V (R4N+I ) in water (Millero, 1971) are plotted against the relative molecular mass of the cation, M+, the intercept at M + = 0 is equated to Ve (I-) (Conway et al., 1966). This procedure has been used to... 

These elements, with five valence electrons, form bonds which are almost exclusively covalent in character. For this reason, it is not generally profitable to invoke an ionic description, however formal, and the definition of an ionic radius is usually of Httle value. An exception might be made in certain phosphorus complexes where an empirical radius may be defined and used in the usual way. 

The trivalent state is the characteristic one for all the lanthanides. They form oxides, M203, which resemble the Ca-Ba group oxides and absorb carbon dioxide and water from the air to form carbonates and hydroxides, respectively. The hydroxides, M(OH)3, are definite compounds, having hexagonal structures, and not merely hydrous oxides. The basicities of the hydroxides decrease with increasing atomic number, as would be expected from the decrease in ionic radius. The hydroxides are precipitated from aqueous solutions by ammonia or dilute alkalis as gelatinous precipitates. They are not amphoteric. 

To a first approximation, the individual ions in the structure of sol d bodies behave as rigid spheres. Each of them, e.g. the univalent Na ion, the divalent Fe ion, etc. possesses a definite ionic radius which is specific to it and persists practically unchanged in the passage from one lattice... 

Definition of the ionic radius as viewed from a hard-spheres ionic crystalline lattice (a) and an electron density contour map (b). [Reproduced from Kittel, C. Introduction to Solid State Physics, 6th ed., John Wiley Sons, Inc New York, 1986. This material is reproduced with permission of John Wiley Sons, Inc.]... 

In Table 1.1 we have collected some atomic and nuclear properties of the halogens, fluorine being included for completeness and comparison. To stress the ambiguity of some properties such as electronegativity and ionic radius we have listed values from different authors - the original articles should be consulted for an account of the underlying criteria and definitions. 

A monatomic ion, like an atom, is a nucleus surrounded by a distribution of electrons. The ionic radius is a measure of the size of the spherical region around the nucleus of an ion within which the electrons are most likely to be found. As for an atomic radius, defining an ionic radius is somewhat arbitrary, because an electron distribution never abruptly ends. However, if we imagine ions to be spheres of definite size, we can obtain their radii from known distances between nnclei in crystals. (These distances can be determined accurately by observing how crystals diffract X rays.) ... 

Air has a refractive index of 1.0003, such that for most purposes, one can replace vacuum by air in the definition, n is thus a measure of the slowing down of a light ray entering a denser medium. Refractive indices are sometimes denoted R.I., which is the inverse of I.R. used for the quite unrelated ionic radius (see p. 359) however, in French both are perfectly inverted with respect to English (i.e., LR. = indice de refraction and R.I. = rayon ionique. 

This argument explains quite well the dependence of Tc on the A-site ion size observed by Hwang et al. [15] and reproduced in Fig. 15. The value of Tp declines sharply when the A-site ionic radius (rA) becomes smaller than 1.25 A (Lao.ySro.s), and saturates at 1.18 A (Pro.yCao.s). Here Hwang et al. used the 9-coordinated ionic radius, and in our definition they correspond to 1.39 A and 1.34 A, respectively. Above (tA) = 1.386 A the system is metallic and Tc... 

Compared with some other atomic parameters, the values of ionic radii have more definite physical meaning. Although the ionic radius changes with coordination number, the change is not very large. 

Within a spherical space of radius a, by definition Qy = 0, so that the value of potential of the ionic atmosphere here is constant and equal to that at point r = a ... 

The definition of the radius of an ion in a crystal as the distance along the bond to the point of minimum electron density is identical with the definition of the radius of an atom in a crystal or molecule that we discuss in the analysis of electron density distributions in Chapter 6. The radius defined in this way does not depend on any assumption about whether the bond is ionic or covalent and is therefore applicable to any atom in a molecule or crystal independently of the covalent or ionic nature of the bond, but it is not constant from one molecule or crystal to another. The almost perfectly circular form of the contours in Figure... 

The first satisfactory definition of crystal radius was given by Tosi (1964) In an ideal ionic crystal where every valence electron is supposed to remain localised on its parent ion, to each ion it can be associated a limit at which the wave function vanishes. The radial extension of the ion along the connection with its first neighbour can be considered as a measure of its dimension in the crystal (crystal radius). This concept is clearly displayed in figure 1.7A, in which the radial electron density distribution curves are shown for Na and Cl ions in NaCl. The nucleus of Cl is located at the origin on the abscissa axis and the nucleus of Na is positioned at the interionic distance experimentally observed for neighboring ions in NaCl. The superimposed radial density functions define an electron density minimum that limits the dimensions or crystal radii of the two ions. We also note that the radial distribution functions for the two ions in the crystal (continuous lines) are not identical to the radial distribution functions for the free ions (dashed lines). 

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