At Dielectric Saturation

In radial fields near an ion, the population of Langevin dipoles in a given interaction range becomes rapidly less with decreasing ion-dipole distance, e.g., in a shell of thickness 5.8 A with a center at 6.3 A there are about 20 superdipoles, with about 5 times as many in the next shell. A less approximate Stirling expression 

0 = 0 and 7t, for hindered rotation in a crystal) and for three-dimensional rotation with n minima (at q = 0, Inin, 2jt/2n, etc., for hindered internal rotation of, e.g., CH3 groups within a molecule rotating in space) have been evaluated.149,150 The eigensolutions for such equations transition from those for a non-degenerate free rotation (when V 0.25kT) to approximately those for a harmonic vibration (when V 0.25kT). From Eqs. (35) and (36), this transition will occur at practically the bulk s0 value at 7.3, 10.3, and 12.6 A from ions with z = 1, 2, and 3 respectively. For z = 1, this is a little beyond the center of the first Continuum shell, while for z = 3, it is closer than the center of the second shell. At the center of the first Continuum shell at 6.3 A for z = 

We therefore arrive at Eq. (53), the classical expression used by Bernal and Fowler16 for the inner sphere, with the addition of a small additional term, which may be regarded as a +TAS term for the breakup of the solvent structure, which can be replaced by an estimate of the energy to break the structure of water. 

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