Diffuse second sphere

These q values are given an estimated uncertainty of 0.5. This can be improved by the use of the modified Horrocks equation determined by Parker and co-workers (24), which takes into account closely diffusing second sphere solvent molecules and proximate N-H bonds in the ligand see Eqs. (2) and (3). 

Phosphonate or carboxylate groups of the ligand are capable of tightly binding water molecules that cannot be treated any more using only translational diffusion. For such cases, a second sphere contribution (r nd) has to be added to the overall proton... 

In some instances it has been possible to use isostructural substitution for the analysis of X-ray diffraction data. The method was first applied by Bol et al. (32) for an investigation of the coordination around Co(II), Ni(II), Zn(II), and Cd(II) in aqueous chloride solutions. The pairs Co(N03)2-Mg(N03)2) Ni(N03)2-Mg(N02)2, Zn(N03)2-Mg(N03)2, and Cd(N03)2-Co(N03)2 were assumed to form isostructural solutions, although the ionic radii are slightly different. The calculated difference RDFs showed two pronounced peaks corresponding to a distinct first and a more diffuse second coordination sphere, which could be analyzed by comparison with theoretical curves. 

The outer-sphere relaxation originates from the translational diffusion of solvent molecules around the metal center, and consists of the outer-sphere relaxation and the second-sphere relaxation. As there is no chemical interaction or electron transfer between the solvent molecules and the metal center, it is more difficult to rationalize. A hard sphere diffusion... 

The particles located upstream are shielding the subsequent particles. As a result, the total diffusion flux on their surfaces decreases monotonically so that I > h> > Ik> ik+i > , and the ratio h/I tends to zero as k - oo. As follows from (4.14.5), the total diffusion flux on the second sphere is nearly half the total diffusion flux on the first sphere, and that on the seventh sphere is less than one-third of that on the first. 

Already Ostwald (6 ) distinguished primary hydration spheres around organic solutes and a second diffuse hydrate sphere. Bungenberg de Jong assumed coacervate formation as combination of... 

Once water molecules are removed, deuterating the central C-H group of the -diketone further increases the quantum yields and lifetimes by approximately 17% in the ternary complexes. Another source of nonradiative deactivation that should not be neglected comes from diffusing solvent molecules in the second sphere which contain high-frequency C-H vibrations. As a consequence, replacement of toluene with carbon tetrachloride leads to a further slight increase in the luminescence lifetimes of [Yb(dbm-6 i)3(phen)] and [Yb(tta-Ji)3(phen)] (table 10). 

Rizkalla and Choppin (1991, 1994) have reviewed the hydration of lanthanide ions. They report that experimentally determined values (by electrophoresis and diffusion) for the hydrated radii of the lanthanides increase from La to Dy but i jparently level off for the heavier lanthanides (fig. 7). Replicate determinations by different authors place the uncertainty on these experimental values at 0.02-0.03 A. The apparent discontinuity near Tb is curious, but is paralleled by the heats and fiee energies of formation of the aquo cations. The similarity suggests that the observed trend represents a real chemical characteristic of the ions, perhaps related to the change in the inner-sphere coordination number or the balance of inner-sphere/second-sphere hydration. A simple analysis of the ions based on these hydrated radii indicates hydration numbers of 12-15 across the series (Lundqvist 1981). David and Fourest (1997) offer a more detailed interpretation that suggests a larger number of waters associated with the lanthanide cations. 

For Gd(III) complexes there are several processes that can contribute to this correlation time. Electronic relaxation (l/Fi e) at the Gd(III) ion, rotational diffusion (1/tr) of the complex, and water exchange in and out of die first (l/tn,) or 2nd (1/Xni ) coordination sphere all create a fluctuating field that can serve to relax the hydrogen nucleus. It is die fastest rate (shortest time constant) that determines the extent of relaxation. For water in the second sphere, the relevant correlation time may be the lifetime of diis water, which may be on the order of tens of picoseconds. Water in the inner sphere typically has a much longer residency time (1-10,000 ns), so the relevant correlation time is usually rotational diffusion or electronic relaxation. 

For a chemical reaction such as combustion to proceed, mixing of the reactants on a molecular scale is necessary. However, molecular diffusion is a very slow process. Dilution of a 10-m diameter sphere of pure hydrocarbons, for instance, down to a flammable composition in its center by molecular diffusion alone takes more than a year. On the other hand, only a few seconds are required for a similar dilution by molecular diffusion of a 1-cm sphere. Thus, dilution by molecular diffusion is most effective on small-scale fluctuations in the composition. These fluctuations are continuously generated by turbulent convective motion. 

The Dunwald—Wagner equation, based on the application of Ficks second law of diffusion into or out of a sphere (radius r) [477], can be written... 

This equation reflects the rate of change with time of the concentration between parallel planes at points x and (x + dx) (which is equal to the difference in flux at the two planes). Fick s second law is vahd for the conditions assmned, namely planes parallel to one another and perpendicular to the direction of diffusion, i.e., conditions of linear diffusion. In contrast, for the case of diffusion toward a spherical electrode (where the lines of flux are not parallel but are perpendicular to segments of the sphere), Fick s second law has the form... 

Consider the specific example of a spherical electrode having the radius a. We shall assume that diffusion to the spherical surface occurs uniformly from all sides (spherical symmetry). Under these conditions it will be convenient to use a spherical coordinate system having its origin in the center of the sphere. Because of this synunetry, then, aU parameters have distributions that are independent of the angle in space and can be described in terms of the single coordinate r (i.e., the distance from the center of the sphere). In this coordinate system. Pick s second diffusion law becomes... 

The first and second integrals have their coordinate systems centered on the catalytic C and noncatalytic N spheres, respectively. The local nonequilibrium average microscopic density field for species a is pa(r) = [Y = 5(r - ( )) The solution of the diffusion equation can be used to estimate this nonequilibrium density, and thus the velocity of the nanodimer can be computed. The simple model yields results in qualitative accord with the MPC dynamics simulations and shows how the nonequilibrium density field produced by reaction, in combination with the different interactions of the B particles with the noncatalytic sphere, leads to directed motion [117],... 

Membrane diffusion illustrates the uses of Fick s first and second laws. We discussed steady diffusion across a film, a membrane with and without aqueous diffusion layers, and the skin. We also discussed the unsteady diffusion across a membrane with and without reaction. The solutions to these diffusion problems should be useful in practical situations encountered in pharmaceutical sciences, such as the development of membrane-based controlled-release dosage forms, selection of packaging materials, and experimental evaluation of absorption potential of new compounds. Diffusion in a cylinder and dissolution of a sphere show the solutions of the differential equations describing diffusion in cylindrical and spherical systems. Convection was discussed in the section on intrinsic dissolution. Thus, this chapter covered fundamental mass transfer equations and their applications in practical situations. 

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