Radii crystal

Calculate the surface energy at 0 K of (100) planes of radon, given that its energy of vaporization is 35 x 10 erg/atom and that the crystal radius of the radon atom is 2.5 A. The crystal structure may be taken to be the same as for other rare gases. You may draw on the results of calculations for other rare gases. 

If the nucleus wets the catalyst well, with 6= 10°, say, then eqn. (7.15) tells us that het IS.lrt,. In other words, if we arrange our 10 atoms as a spherical cap on a good catalyst we get a much bigger crystal radius than if we arrange them as a sphere. And, as Fig. 7.4 explains, this means that heterogeneous nucleation always "wins" over... 

Fig. 7.4. Heterogeneous nucleation takes place at higher temperatures because the maximum random fluctuation of 10 atoms gives a bigger crystal radius if the atoms are arranged as a spherical cap.

From this equation we can calculate the actual crystal radius (for use in normal sodium chloride type crystals) from Ru the univalent crystal radius. A knowledge of the repulsion exponent n is needed, however. This can be derived from the experimental measurement of the compressibility of crystals. Bom11 and Herzfeld,12 give the values in Table III, obtained in this way. 

The radii found in this way are given in Table II, the univalent radii being also included, in parentheses. The dependence of the univalent crystal radius and the crystal radius on the atomic number is shown graphically in Fig. 3.14 The effect of the valence in causing the crystal radius to deviate from the regular dependence on the atomic number shown by the univalent crystal radius is clearly evident. 

Fig. 3.—The crystal radius (solid circles) and the univalent crystal radius (open circles) for a number of ions.

In deriving theoretical values for inter-ionic distances in ionic crystals the sum of the univalent crystal radii for the two ions should be taken, and corrected by means of Equation 13, with z given a value dependent on the ratio of the Coulomb energy of the crystal to that of a univalent sodium chloride type crystal. Thus, for fluorite the sum of the univalent crystal radii of calcium ion and fluoride ion would be used, corrected by Equation 13 with z placed equal to y/2, for the Coulomb energy of the fluorite crystal (per ion) is just twice that of the univalent sodium chloride structure. This procedure leads to the result 1.34 A. (the experimental distance is 1.36 A.). However, usually it is permissible to use the sodium chloride crystal radius for each ion, that is, to put z = 2 for the calcium... 

It is to be expected that the relative values of the univalent crystal radius would be of significance with respect to physical properties involving atomic sizes. That this is true for the viscosity of the rare gases is seen from the radii evaluated by Herzfeld (Ref. 12, p. 436) from viscosity data He, 0.04 (0.93) Ne, 1.18 (1.12) Ar, 1.49 (1.54) Kr, 1.62 (1.69) Xe, 1.77 (1.90). (The values in parentheses are the univalent crystal radii.)... 

The effect of deformation is shown in the sulfide, selenide and telluride of lead. Especially interesting is the decrease in the crystal radius in the series Mn++, Fe++, Co++, Ni++ there must then come an increase when the shell is completed, at Zn++, with the radius 0.74 A. 

We have accordingly shown that for values of the ratio of the crystal radius of the cation to that of the anion greater than 0.65 the fluorite structure is stable for values less than 0.65 the rutile structure is stable. 

In this discussion, two mutually canceling simplifications have been made. For the transition value of the radius ratio the phenomenon of double repulsion causes the inter-atomic distances in fluorite type crystals to be increased somewhat, so that R is equal to /3Rx-5, where i has a value of about 1.05 (found experimentally in strontium chloride). Double repulsion is not operative in rutile type crystals, for which R = i M + Rx- From these equations the transition ratio is found to be (4.80/5.04)- /3i — 1 = 0.73, for t = 1.05 that is, it is increased 12%. But Ru and Rx in these equations are not the crystal radii, which we have used above, but are the univalent crystal radii multiplied by the constant of Equation 13 with z placed equal to /2, for M++X2. Hence the univalent crystal radius ratio should be used instead of the crystal radius ratio, which is about 17% smaller (for strontium chloride). Because of its simpler nature the treatment in the text has been presented it is to be emphasized that the complete agreement with the theoretical transition ratio found in Table XVII is possibly to some extent accidental, for perturbing influences might cause the transition to occur for values a few per cent, higher or lower. 

Compounds are classified by AR = Ra-Rc) as shown in the legend, where Ra and Rc are the crystal radius of a cation and an anion, respectively. 

Calculation of the Solvation Energy from Experimental Data The solvation energies of individual ions can be calculated from experimental data for the solvation energies of electrolytes when certain assumptions are made. If it is assumed that an ion s solvation energy depends only on its crystal radius (as assumed in Bom s model), these energies should be the same for ions K+ and F , which have similar values of these radii (0.133 0.002nm). It follows that in aqueous solutions, K+ = F- = = 414.0 kJ/mol. With the aid of these values we can now determine... 

The values of hj for different ions are between 0 and 15 (see Table 7.2). As a rule it is found that the solvation number will be larger the smaller the true (crystal) radius of the ion. Hence, the overall (effective) sizes of different hydrated ions tend to become similar. This is why different ions in solution have similar values of mobilities or diffusion coefficients. The solvation numbers of cations (which are relatively small) are usually higher than those of anions. Yet for large cations, of the type of N(C4H9)4, the hydration number is zero. 

The conductivities of melts, in contrast to those of aqueous solutions, increase with decreasing crystal radius of the anions and cations, since the leveling effect of the solvation sheaths is absent and ion jumps are easier when the radius is small. In melts constituting mixtures of two salts, positive or negative deviations from additivity are often observed for the values of conductivity (and also for many other properties). These deviations arise for two reasons a change in hole size and the formation of new types of mixed ionic aggregates. 

The effects of the anions (i.e., their specific adsorbabilities) increase in the order F < Cr < Br < I . This trend is due to the fact that the solvation energy decreases with increasing crystal radius as one goes from F to I , and the transfer of the ions to the inner Helmholtz plane is facilitated accordingly. The opposite picture is seen for surface-active cations (e.g., [N(C4H5)4]+) the descending branch of the ECC is depressed, and the PZC shifts in the positive direction. 

Micropore mass transfer resistance of zeoUte crystals is quantified in units of time by r /Dc, where is the crystal radius and Dc is the intracrystalline diffusivity. In addition to micropore resistance, zeolitic catalysts may offer another type of resistance to mass transfer, that is resistance related to transport through the surface barrier at the outer layer of the zeoHte crystal. Finally, there is at least one additional resistance due to mass transfer, this time in mesopores and macropores Rp/Dp. Here Rp is the radius of the catalyst pellet and Dp is the effective mesopore and macropore diffusivity in the catalyst pellet [18]. 

While microscopic techniques like PFG NMR and QENS measure diffusion paths that are no longer than dimensions of individual crystallites, macroscopic measurements like zero length column (ZLC) and Fourrier Transform infrared (FTIR) cover beds of zeolite crystals [18, 23]. In the case of the popular ZLC technique, desorption rate is measured from a small sample (thin layer, placed between two porous sinter discs) of previously equilibrated adsorbent subjected to a step change in the partial pressure of the sorbate. The slope of the semi-log plot of sorbate concentration versus time under an inert carrier stream then gives D/R. Provided micropore resistance dominates all other mass transfer resistances, D becomes equal to intracrystalline diffusivity while R is the crystal radius. It has been reported that the presence of other mass transfer resistances have been the most common cause of the discrepancies among intracrystaUine diffusivities measured by various techniques [18]. 

One is translational diffusion, which produces a variation of the distance between the particle and the water molecule. The translation correlation time is the time for the water molecule to diffuse a crystal radius ... 

Id = r jD where r is the crystal radius and D the water diffusion coefficient. 

For USPIO particles containing only one nanomagnet per particle, the main parameters determining the relaxivity are the crystal radius, the specific magnetization and the anisotropy energy. Indeed, the high field dispersion is determined by the translational correlation time t. ... 

Dependence of the Crystal Radius and of the Magnetization Obtained by Relaxometry AND Magnetometry with the [Fe ]/[Fe ] Ratio... 

The volume properties of crystalline mixtures must be related to the crystal chemical properties of the various cations that occupy the nonequivalent lattice sites in variable proportions. This is particularly true for olivines, in which the relatively rigid [Si04] groups are isolated by Ml and M2 sites with distorted octahedral symmetry. To link the various interionic distances to the properties of cations, the concept of ionic radius is insufficient it is preferable to adopt the concept of crystal radius (Tosi, 1964 see section 1.9). This concept, as we have already noted, is associated with the radial extension of the ion in conjunction with its neighboring atoms. Experimental electron density maps for olivines (Fujino et al., 1981) delineate well-defined minima (cf figure 1.7) marking the maximum radial extension (rn, ,x) of the neighboring ions ... 

In light of equations 5.18 and 5.19, a relationship can be derived between ionization potential (/) and crystal radius (Della Giusta et ah, 1990) ... 

The solvation energy described by the Born equation is essentially electrostatic in nature. Born equations 8.116 and 8.120 are in fact similar to the Born-Lande equation (1.67) used to define the electrostatic potential in a crystal (see section 1.12.1). In hght of this analogy, the effective electrostatic radius of an ion in solution r j assumes the same significance as the equilibrium distance in the Born-Lande equation. We may thus expect a close analogy between the crystal radius of an ion and the effective electrostatic radius of the same ion in solution. 

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). 

The crystal radius thus has local validity in reference to a given crystal structure. This fact gives rise to a certain amount of confusion in current nomenclature, and what it is commonly referred to as crystal radius in the various tabulations is in fact a mean value, independent of the type of structure (see section 1.11.1). The crystal radius in the sense of Tosi (1964) is commonly defined as effective distribution radius (EDR). The example given in figure 1.7B shows radial electron density distribution curves for Mg, Ni, Co, Fe, and Mn on the M1 site in olivine (orthorhombic orthosilicate) and the corresponding EDR radii located by Fujino et al. (1981) on the electron density minima. 

This distinction between electron-cloud radii and structural radii is then used to refine the system of ionic radii due to Pauling and Goldschmidt. Some further examples of anion-anion contact are discussed, and a value deduced for the crystal radius of the hydride anion. These cases of anion-anion contact argue for the Pauling tradition and against the new electron-density-minimum (EDM) radii. 

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