Onsager function

Khokhlov and Semenov derived analytical expressions for a(N) from Eqs. (12)—(16) in the asymptotic limits of N 1 and N l, assuming no special form for U(a). Furthermore, by choosing the Onsager function [2] as the trial function for f(a), they expressed cr(N) as functions of N and a in these asymptotic limits. Here a is the parameter representing the degree of orientation, which appears in the trial function used, i.e.,... 

The solvent polarity function F(D) = (1 — 1 /D) having the same saturation property as the Debye and Onsager functions. As D goes to infinity all three functions approach the limit of 1. 

Figure 3.57 Solvatochromic plot of the first absorption band of a betaine dye. The transition corresponds to a CT from 0 to the aromatic system. Triangles are protic solvents. E is in kcal moE1 and f(D) is the Onsager function...

The Onsager function is dependent on the refractive index of the solvent (n). For supercritical C02 the refractive index is dependent on the fluid density and can be calculated from the Lorentz-Lorenz refraction equation ... 

Figure 15-1. B3LYP interaction energies computed for each labeled proton donor with water as acceptor. Onsager function F0 is equal to (e-l)/(e+2) where e refers to dielectric constant of the medium...

The specific construction of the Rayleigh Onsager function (2.6) as well as the Ljk independence of internal forces X gives... 

This conclusion is a principal statement of the I. Prigogine theorem (1947, the Nobel Prize winner in 1977). It also is essential in view of the positively determined Rayleigh Onsager function that the spontaneous evolution of the system to its stationary state can be accompanied by only a monotonous decrease in P and, as a result, in that is,... 

The simplest case is to find the Lyapunov function for systems with an arbitrary set of the intermediate linear chemical transformations. In prox imity to thermodynamic equilibrium, this function becomes identical to the Rayleigh Onsager functional and thus relates to the principle of the minimal rate of entropy production. 

Here, index i and parameter are substituted for by index a and parame ter Ao(, which is always allowed for monomolecular processes. Thus, the functional O derivative with respect to thermodynamic rush of an inter mediate is proportional to the rate of this intermediate concentration changes everywhere, even far from thermodynamic equihbrium. For this reason, in the state stationary with respect to the intermediate concentra tion, this derivative turns zero (cf the case of the Rayleigh Onsager functional). 

Classifying variables into fast or slow is a typical approach in chemical kinetics to apply the method of (quasi)stationary concentrations, which allows the initial set of differential equations to be largely reduced. In the chemically reactive systems near thermodynamic equihbrium, this means that the subsystem of the intermediates reaches (owing to quickly changing variables) the stationary state with the minimal rate of entropy production (the Rayleigh Onsager functional). In other words, the subsys tern of the intermediates becomes here a subsystem of internal variables. 

Finally, a multiparameter correlation of the meager solvent effect data indicates that solvent polarity as measured by the Kirkwood-Onsager function, (e — l)/(2e -h 1), and hydrogen bonding as measured by the free energy of transfer of chloride ion between the solvents and gas phase both contribute to the modest solvent effect observed. The dominant effect is appropriate to each solvent. 

Ray et al. [87] have studied solvatochromism of POT using mixed solvents of NMP and CH2CI2 and have plotted the solvent effect against the mole fraction of the polar solvent, and it revealed preferential solvation of POT by NMP. The Onsager function (/(e) and/(n ) are additive properties in the case of ideal mixed solvents. Therefore, the polarity function for the mixture can be written as... 

The temperature dependence of f (n ) arises from the changes in the solvent density with temperature. Since the change in the dielectric constant with temperature is small, the refractive index-dependent term in the above equation cannot be neglected as was done for solvatochromism in equation (13.4). This is particularly important for polar solvents, since the Onsager function /(e) is less sensitive at higher values of T and hence the temperature dependence of /(e) is small for polar solvents. 

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