Onsager equation/radius

HPLC = high performance liquid chromatography see chromatography HOckel-Onsager equation see electrophoretic mobility hydration 1.2.58, 1.5.3, 3.121, table 3.7 (see also solvation) hydration number 1.5.50 hydraulic radius 1.6.50, 1.84 hydrod3mamlc radius, layer thickness 1.7.50, 5.61 hydrod)mamics 1.6.1,... 

These are general equations that allow calculating solvatochromic shifts if we know electric characteristics of the solute (dipole moments in the ground and excited states, pg and /Onsager radius a, and the function of interactions/(fio, o). 

Equation (9.2) shows the effect of the reaction radius rl on the escape probability, which, remarkably, is free of the diffusion coefficient. Normally rc r which reduces Eq. (9.2) to the celebrated Onsager formula 0 = exp(- r /rg) as given by Eq. (9.1). 

R is the radius of the cavity, p and a are the dipole moment and polarizability of the solute, and s the dielectric constant of the solvent. Equation (35) does address the polarization of the solute molecule by the reaction field, although not carrying this to self-consistency. (It is interesting that Onsager s paper, the sixth-most-cited in the history of the Journal of the American Chemical Society, was rejected by the Physikalische Zeitshrift, to which it had initially been submitted.)89... 

Fig. 4.5. Effective radii of the diffusion-controlled tunnelling recombination for the Coulomb attraction (a) and repulsion (b) (after [65]). Curve 1 and 2 are results of computer calculations with parameters cr = 107s l, r = 20 A, Rd = 4 A and cr = 10l4s l, r = 2 A, Rd = 4 A respectively. L - the Onsager radius, equation (3.2.55), Ro - radius of strong tunnelling recombination, equation (4.3.7).

The value of the parameter L entering equation (6.4.1) defines whether the Coulomb attraction or recombination is predominant as L effective recombination sphere equals the Onsager radius). 

For CS and CR processes, an alternative is provided by the Onsager model [46] of a point dipole in a sphere (of mean radius rD/A). In the limiting case of CS (CR), the initial (final) state dipole moment (/z) is zero, and the shift in dipole (Ap,) is given by the final (initial) p value. This leads to the Lippert-Mataga (LM) dipolar analog of Equation (3.93) [47]... 

The expression most commonly used in fluorescence spectroscopy is, however, the somewhat simphfied Eq. (6-5b), first developed by Lippert [47, 488] and Mataga [14, 489]. It is based on Onsager s reaction-field theory, which assumes that the fluorophore is a point dipole residing in the center of a spherical cavity with radius a in a homogeneous and isotropic dielectric with relative permittivity e,. The so-called Lippert-Mataga equation is as follows ... 

In the harmonic approximation in which a parabolic well near the equilibrium bond length is assumed, the frequencies will not be altered, but inclusion of anharmonic (cubic and higher order) terms in the intraionic potential function will further modify the effective force constant of the impurity ion. F was introduced via an expression by Bauer and Magat [26] and modified by Maki and Decius [27] based on Onsager s model of a homogeneous solvent of dielectric constant e with the solute contained in a spherical cavity of radius a. The frequency shift equation is given by... 

The cavity size in the Bom/Onsager/Kirkwood models strongly influences the calculated stabilization. Unfortunately, there is no consensus on how to choose the cavity radius. In some cases, the molecular volume is calculated from the experimental density of the solvent and the cavity radius is defined by equating the cavity volume to the molecular volume. Alternatively, the cavity size may be derived from the (experimental) dielectric constant and the calculated dipole moment and polarizability. In any case, the underlying assumption of these models is that the molecule is roughly spherical or ellipsoidal, which is only generally true for small compact molecules. 

The other model for the ionic friction concerns the dielectric response of solvent to the solute perturbation. When an ion is fixed in polar solvent, the solvent is polarized according to the electrostatic field from the ion. If the ion is displaced, the solvent polarization is not in equilibrium with a new position of the ion, and the relaxation of the polarization should take place in the solvent. The energy dissipation associated with this relaxation process may be identified as an extra friction. The extra friction, called the dielectric friction, decreases with increasing ionic radius, thereby, with decreasing electrostatic field from the ion. The dielectric friction model developed by Born [66], Fuoss [67], Boyd [68] and Zwanzig [69, 70] has taken a complete theoretical form due to the work by Hubbard and Onsager [71, 72] who proposed a set of continuum electrohydrodynamic equations in which the electrostatic as well as hydrodynamic strains are incorporated. 

The Theory of Onsager (5) In the case of an nonpolarizable solute, the model is quite simple The solute molecule A is assumed to be a point dipole (tx located at the center of a cavity of radius a. The potential created by the dipole and its surroundings has to satisfy Laplace s equation /... 

Notably, the use of the macroscopic dielectric constant s = Sq in the last formula is justified only when the lifetime of the solute molecule in a given (v-th) state is much longer than the rotational-vibrational relaxation time of the solvent at given temperature. This is not a valid assumption in the case of the Franck-Condon states, which have the lifetime much shorter than the rotational-vibrational relaxation time of the solvent. Therefore, the solvent is only partially relaxed for these states and the corresponding reaction field is characterized by the dielectric constant at infinite frequency of external electric field, 8 . By inserting the expression for the reaction field [11.1.36] into the equation [11.1.18] and assuming that the static polarizability of the solute molecule is approximately equal to the one third of the cube of Onsager s cavity radius... 

The theory of Onsager-Bottcher is an attempt to improve the theory of the optical behaviour of mixtures, taking the influence of the internal field into consideration. Substituting the observed refractive indices in the somewhat more complicated equations of Onsager-BoTTCHER the average polarisability of the glucose residue and the radius r of the same could be calculated from the observations ... 

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