Debye-Onsager model

Debye-Onsager model for C(t) and the longitudinal relaxation time Tj... 

An Evaluation of the Debye-Onsager Model. The simplest treatment for solvation dynamics is the Debye-Onsager model which we reviewed in Section II.A. It assumes that the solvent (i) is well modeled as a uniform dielectric continuum and (ii) has a single relaxation time (i.e., the solvent is a Debye solvent ) td (Eq. (18)). The model predicts that C(t) should be a single... 

Ignoring the potential limitations of the dielectric data, we can evaluate the Debye-Onsager model for a number of apparently roughly Debye solvents, like propylene carbonate, the alkyl nitriles, the alkyl acetates, and other solvents. First of all, C( ) is often strongly nonmonoexponential, in contradiction to the theoretical prediction. Second, the observed average solvation time is often much different from xt. 

A number of theoretical models for solvation dynamics that go beyond the simple Debye Onsager model have recently been developed. The simplest is an extension of Onsager model to include solvents with a non-Debye like (dielectric continuum and the probe can be represented by a spherical cavity. Newer theories allow for nonspherical probes [46], a nonuniform dielectric medium [45], a structured solvent represented by the mean spherical approximation [38-43], and other approaches (see below). Some of these are discussed in this section. Attempts are made where possible to emphasize the comparison between theory and experiment. 

Finally, it must be recalled that the transport properties of any material are strongly dependent on the molecular or ionic interactions, and that the dynamics of each entity are narrowly correlated with the neighboring particles. This is the main reason why the theoretical treatment of these processes often shows similarities with models used for thermodynamic properties. The most classical example is the treatment of dilute electrolyte solutions by the Debye-Hiickel equation for thermodynamics and by the Debye-Onsager equation for conductivity. 

As soon as the concentration of the solute becomes finite, the coulombic forces between the ions begin to play a role and we obtain both the well-known relaxation effect and an electrophoretic effect in the expression for the conductivity. In Section V, we first briefly recall the semi-phenomenological theory of Debye-Onsager-Falkenhagen, and we then show how a combination of the ideas developed in the previous sections, namely the treatment of long-range forces as given in Section III and the Brownian model of Section IV, allows us to study various microscopic... 

The Onsager Model in the Non-linear Electric Field Effect. The non-linear dectric field theory of Debye disagrees with most of the measurements of Ac in liquids and thdr solutions. To Van Vleck is due the atten t to reduce the discrepandes by applying Onsager s local electric field model in the treatment of the field effect. Aiming at simplidty, he considered only the influence of reorientation of ri d (not polarizable) electric dipoles p,... 

Fig. 6.10 Plot of the equivalent conductance of NaCl in water according to equation (6.9.1) against the square root of the ionic strength at 25°C. The solid curve shows the prediction of the Debye-Onsager equation with a = 0.4 nm, and the straight line, the prediction of this model in the limit that ion size effects may be neglected.

In this chapter we present a simple model calculation that demonstrates how this cooperative motion affects the scattering spectrum. Our approach is based on the Debye-Onsager treatment of ion transport (see Falkenhagen, 1934 Stephen, 1971). This is our first discussion of cooperative effects in light scattering. In Chapter 13 this problem is reconsidered in the context of the general theory of nonequilibrium thermodynamics. 

SASA), a concept introduced by Lee and Richards [9], and the electrostatic free energy contribution on the basis of the Poisson-Boltzmann (PB) equation of macroscopic electrostatics, an idea that goes back to Born [10], Debye and Htickel [11], Kirkwood [12], and Onsager [13]. The combination of these two approximations forms the SASA/PB implicit solvent model. In the next section we analyze the microscopic significance of the nonpolar and electrostatic free energy contributions and describe the SASA/PB implicit solvent model. 

The physical meaning of the relationship described in the previous subsection becomes apparent when we consider the popular special case of the Onsager cavity model that arises if we assume that the solvent s dielectric properties are well described by a Debye form. 

Onsager Theory for C(t) for Non-Debye Solvents. Generally solvents have more complex dielectric responses than described by the Debye equation (Eq. (18)). To obtain the time dependence of the reaction field R from Eqs. (12, (15), (16) and (7) an appropriate model for dielectric behavior of a specific liquid should be employed. One of the most common dielectric relaxation is given by the Debye-type form, which is applicable to normal alcohols. 

A theoretical approach for explaining the relationship between S and the characteristics of the electrolyte was provided by Onsager on the basis of the model of ions plus ionic cloud developed in the Debye-Hiickel theory, obtaining [4]... 

Onsager 0 equations (Section 5.10), one can extract the dipole moments of polar molecules and the polarizability of any solute molecule. One needs a capacitance cell whose electrodes are as close to each other as practical (for higher capacitances) and reasonable solubilities. If the shape of the solute is very different from the sphere used in the Debye model, then the ellipsoidal cavity has been treated theoretically [13] and applied to hypsochromism [14]. 

In dilute electrolyte solutions ion-ion interaction as function of electrolyte concentration is fully explained by the Debye-Hiickel-Onsager theory and its further development. The contribution of ion solvation is noticed, if, for instance, the mobilities at infinite dilution of an ion in different solvent media or as function of ionic radii as considered. Up till now the calculation of that dependence has been only rather approximateAn improvement is quite probable, though, theoretically very involved if the solvent is not regarded as a continuum, but the number and arrangement of solvent molecules in the primary solvation shell of an ion is taken into consideration. Also the lifetime of molecules in the solvation shell must be known. Beyond this region a continuum model of ion-solvent interaction may be sufficient to account for the contributions of solvent molecules in the second or third sphere. 

The application of Blum s theory to experiment is unexpectedly impressive it can even represent conductance up to 1 mol dm . Figure 4.96 shows experimental data and both theories—Blum s theory and the Debye-Hiickel-Onsager first approximation. What is so remarkable is that the Blum equations are able to show excellent agreement with experiment without taking into account the solvated state of the ion, as in Lee and Wheaton s model. However, it is noteworthy that Blum stops his comparison with experimental data at 1.0 M. 

The correction factor,/, relates the actual mobility of a fully charged particle at the ionic strength under the experimental conditions to the absolute mobility. It takes ionic interactions into account and is derived for not-too-concentrated solutions by the theory of Debye-Htickel-Onsager using the model of an ionic cloud around a given central ion. It depends, in a com-... 

In chapter 3, it was shown that the Debye-Hiickel theory for ion-ion interactions is able to account for solution non-ideality in very dilute systems. The same model forms the basis for understanding the concentration dependence of the conductance observed for strong electrolytes. Thus, Onsager [9] showed in 1927 that the limiting conductance law for 1-1 electrolytes has the form... 

In summary, Onsager s extension of the Debye-Hiickel theory to the nonequilibrium properties of electrolyte solutions provides a valuable tool for deriving single ion properties in electrolyte solutions. Examination of the large body of experimental data for aqueous electrolyte solutions helped confirm the model for a strong electrolyte. In more recent years, these studies have been extended to non-aqueous solutions. Results in these systems are discussed in the following section. 

The early conductance theories given by Debye and Hiickel in 1926, Onsager in 1927 and Fuoss and Onsager in 1932 used a model which assumed all the postulates of the Debye-Hiickel theory (see Section 10.3). The factors which have to be considered in addition are the effects of the asymmetric ionic atmosphere, i.e. relaxation and electrophoresis, and viscous drag due to the frictional effects of the solvent on the movement of an ion under an applied external field. These effects result in a decreased ionic velocity and decreased ionic molar conductivity and become greater as the concentration increases. 

We see that Onsager s formula, which is derived from an oversimplified model, usually leads to values of ft correct to about debye. 

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