Ionic radial distribution function

Figure 4.1-11 The EXAFS data and pseudo-radial distribution functions of Co(ll) in (a) basic and (b) acidic chloroaluminate ionic liquid. Reproduced from reference 46 with permission.

The ordering of the anions in bmimX ionic liquids has also been suggested by our recent large-angle x-ray scattering experiment on liquid bmimi [23]. Figure 13 shows a differential radial distribution function obtained for liquid bmimi at room temperature. Clear peaks in the radial distribution curve are... 

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 first method comprises theories based on a statistical description of ionic liquids in a very rigorous way using radial distribution functions obtained from X-ray diffraction. These are theories based on the principle of corresponding states. 

In cluster calculations, an element essential in solution calculations is missing. Thus, intrinsically, gas-phase cluster calculations cannot allow for ionic movement. Such calculations can give rise to average coordination numbers and radial distribution functions, but cannot account for the effect of ions jumping from place to place. Since one important aspect of solvation phenomena is the solvation number (which is intrinsically dependent on ions moving), this is a serious weakness. 

The structural information at an atomic level is essential for understanding the various properties of supercooled and glassy solutions. X-ray and neutron diffraction enables us to obtain direct structure information (bond distance and coordination number) of ionic solutions in terms of the radial distribution function. In the case of aqueous lithium halide solutions. X-ray diffraction data are dominated by halide-oxygen, halide-oxygen, and oxygen-oxygen interactions. On the contrary, neutron isotopic substitution... 

At the same conditions of density and temperature, the smallest cation (Na" ") has larger dynamic hydration number than K" " and Rb", as found experimentally from correlations of conductance measurements. The trend holds at all subcritical conditions. This behavior is opposite to that of static coordination numbers calculated from time-averaged radial distribution functions, which give increasing coordination numbers when the ionic radius increases. Another interesting feature of the hydration numbers is given by their temperature dependence. For all ions at 573 K, dynamic hydration numbers (even when lower than their corresponding coordination numbers) are approximately the same as they are at ambient conditions. The same characteristic is observed at supercritical conditions, as illustrated in Table 12. 

Fig. 1 Diagrams depicting a a layer of a cubic sodium chloride crystal b a monoclinic 1,3-dimethylimidazolium chloride ionic-liquid crystal c two radial distribution functions (RDFs) in liquid l-dodecyl-3-methylimidazolium hexafluorophosphate. Anions and cations are depicted in red and blue. In the cases of b and c the blue circles represent the centroid of the imidazolium rings of the cations. The alternating sequences of red and blue circles in a and b as well as the two curves in phase opposition in c clearly indicate the existence and nature of the polar networks in ionic condensed phases...

Fig. 9 Pair-pair radial distribution functions of different solutes in different ionic liquid solutions. Solvents A l-butyl-3-methylimidazolium hexafluorophos-phate B l-butyl-3-methylimidazolium bis(trifluoromethanesulfonyl)imide C tri-hexyl(tetradecyl)phosphonium bis(trifluoromethanesulfonyl)imide. Solutes ...

This distribution function has already appeared in the Debye-Hiickel theory (Equation 10.22) when the Maxwell-Boltzmann distribution was used to describe the distribution of the ions of the ionic atmosphere around the central j ion (see Figure 10.16). Because the distribution function is given in terms of the distance between an ion of the ionic atmosphere and the central reference j ion it is termed a radial distribution function. This reflects the spherical symmetry assumed in the Debye-Hiickel theory. 

---