Spherical ion

An alternative approach is to consider ions of charge z e accelerated by the electric field strengtii, E, being subject to a frictional force, Kj, that increases with velocity, v, and is given, for simple spherical ions of... 

Consider an alchemical transformation of a particle in water, where the particle s charge is changed from 0 to i) (e.g., neon sodium q = ). Let the transformation be performed first with the particle in a spherical water droplet of radius R (formed of explicit water molecules), and let the droplet then be transferred into bulk continuum water. From dielectric continuum theory, the transfer free energy is just the Born free energy to transfer a spherical ion of charge q and radius R into a continuum with the dielectric constant e of water ... 

R is the distance parameter, defining the upper limit of ion association. For spherical ions forming contact ion pairs it is simply the sum of the crystallographic radii of the ions a — a+ + a for solvent-shared and solvent-separated ion pairs it equals a + s or a + 2s respectively, where s is... 

The structures determined for hematite and corundum show that these crystals consist of a compact arrangement of approximately, but not exactly, spherical ions of oxygen and of iron or aluminum, held together by inter-ionic forces which are prob- atoms in the units of structure of ably electrostatic in nature. No evidence hematite and corundum small cir-... 

Within the hydration process in Eqn. (8.8), a spherical ion becomes a (hydrated) octahedral ion, [M(H20)6]. Part of the Coulomb energy of the free ion concerns repulsion and exchange terms within the d" configuration. This is replaced by equivalent repulsion and exchange terms within the configuration. Let us estimate the trends in these quantities separately. 

Bjerrum considered the case of spherical ions in a solvent of dielectric constant e. The probability of finding two ions of opposite charge at a distance A from each other is calculated from the number of ions surrounding a central ion of opposite charge in a spherical shell of thickness dA and radius A. This probability, fV(A), is given by... 

The structure types discussed so far have a favorable arrangement of cations and anions and are well suited for ionic compounds consisting of spherical ions. However, their occurrence is by no means restricted to ionic compounds. The majority of their representatives are found among compounds with considerable covalent bonding and among intermetallic compounds. 

When three different kinds of spherical ions are present, their relative sizes are also an important factor that controls the stability of a structure. The PbFCl type is an example having anions packed with different densities according to their sizes. As shown in Fig. 7.5, the Cl- ions form a layer with a square pattern. On top of that there is a layer of F ions, also with a square pattern, but rotated through 45°. The F ions are situated above the edges of the squares of the Cl- layer (dotted line in Fig. 7.5). With this arrangement the F -F distances are smaller by a factor of 0.707 (= /2) than the CP-CP distances this matches the ionic radius ratio of rF-/rcl- = 0.73. An F layer contains twice as many ions as a CP layer. Every Pb2+ ion is located in an antiprism having as vertices four F and four... 

The structures of ionic compounds comprising complex ions can in many cases be derived from the structures of simple ionic compounds. A spherical ion is substituted by the complex ion and the crystal lattice is distorted in a manner adequate to account for the shape of this ion. 

Figure 2.4 The dipole moment of a hypothetical purely ionic molecule with spherical ions.

Figure 2.5 (a) Hypothetical ionic molecule with spherical ions and the corresponding charge transfer moment, (b) In a real molecule the ions are polarized leading atomic dipoles in each atom that oppose the charge transfer moment. 

Equation (15) permits a straightforward analysis of dielectric continuum models of hydration that have become popular in recent decades. The dielectric model, also called the Bom approximation, for the hydration free energy of a spherical ion of radius R with a charge q at its center is... 

The classical Born expression for the polarization free energy of a spherical ion of net charge q can be written as82 ... 

The Debye-FIuckel theory for association of spherical ions in a medium of dielectric constant Dr posits that the electrostatic potential energy of interaction between oppositely charged ions is... 

Table 1 Changes in hypothetical potential energies of attraction of oppositely charged spherical ions based on reductions in dielectric constant in passing from water to methanol and ethanol, and the computed changes in binding constant assuming the changes in potential energy are translated into free energies of binding3...

The dielectric displacement must be calculated from electrostatics for a reactant in front of a metal surface the image force has to be considered. For the simple case of a spherical ion in front of a metal electrode experiencing the full image interaction, a straightforward calculation gives ... 

From Born s formula (cf. Problem 6.4) derive an expression for the difference in the energy of solvation of a spherical ion in two solvents with different dielectric constants. 

The direct access to the electrical-energetic properties of an ion-in-solution which polarography and related electro-analytical techniques seem to offer, has invited many attempts to interpret the results in terms of fundamental energetic quantities, such as ionization potentials and solvation enthalpies. An early and seminal analysis by Case etal., [16] was followed up by an extension of the theory to various aromatic cations by Kothe et al. [17]. They attempted the absolute calculation of the solvation enthalpies of cations, molecules, and anions of the triphenylmethyl series, and our Equations (4) and (6) are derived by implicit arguments closely related to theirs, but we have preferred not to follow their attempts at absolute calculations. Such calculations are inevitably beset by a lack of data (in this instance especially the ionization energies of the radicals) and by the need for approximations of various kinds. For example, Kothe et al., attempted to calculate the electrical contribution to the solvation enthalpy by Born s equation, applicable to an isolated spherical ion, uninhibited by the fact that they then combined it with half-wave potentials obtained for planar ions at high ionic strength. 

In the present context P+n A might be poly(iso-butyl)+ A1C14". For the simple case of spherical ions, the dissociation constant, KD, of ion-pairs is governed by the Bjerrum-Fuoss Equation (13) ... 

The Lattice Energy of a Crystal Consisting of Spherical Ions... 

Equation 6.15 is valid only for rigid, spherical ions. In addition, it originates from Equation 6.14, which is applicable only if the electric field at the ion is due to the applied electric field only, undisturbed by the effects of the other ions in solution. Consequently, Equation 6.15 ignores other forces originating in the counterion atmosphere, leaving the influence of the medium on the mobility... 

This form of the equation is valid for linear molecules and symmetric rotors (for which I corresponds to reorientation of the symmetry axis) and can be used for nondipolar solvents. A somewhat more complicated expression would hold in the absence of axial symmetry. Maroncelli et al. estimated the value tti for AP corresponding to a charge shift of a spherical ion in a continuum model of a polar solvent of dielectric constant s and showed that it increases with increasing solvent polarity and works well when tti is significantly larger than one. 

Where Dj Is the diffusion coefficient of species 1 In the continuous medium with dielectric constant D and viscosity n. With reasonably good assumption of spherical Ions of equal radii, and using the Stokes-Einsteln relation for the diffusion coefficients these relations reduce to (since In low polar media e2DkTa < 1) ... 

Further, ions are not hard, billiard ball like spheres. Since the wave functions that describe the electronic distribution in an atom or ion do not suddenly drop to zero amplitude at some particular radius, we must consider the surfaces of our supposedly spherical ions to be somewhat fuzzy. A more subtle complication is that the apparent radius of an ion increases (typically by some 6 pm for each increment) whenever the coordination number increases. Shannon10 has compiled a comprehensive set of ionic radii that take this into account. Selected Shannon-type ionic radii are given in Appendix F these are based on a radius for O2- of 140 pm for six coordination, which is close to the traditionally accepted value, whereas Shannon takes the reference value as 126 pm on the grounds that it gives more realistic ionic sizes. For most purposes, this distinction does not mat-... 

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