Solvent Debye relaxation time

The pre-exponential factor in this case includes the solvent longitudinal relaxation time tl, which will be discussed further on when the recent works concerned with the role of the solvent will be considered. This longitudinal relaxation time is related to the usual Debye relaxation time according to... 

This longitudinal relaxation time differs from the usual Debye relaxation time by a factor which depends on the static and optical dielectric constants of the solvent this is based on the fact that the first solvent shell is subjected to the unscreened electric field of the ionic or dipolar solute molecule, whereas in a macroscopic measurement the external field is reduced by the screening effect of the dielectric [73]. 

Recent theoretical treatments, however, suggest instead that the dynamics of solvent reorganization can play an important and even dominant role in determining vn, at least when the inner-shell barrier is relatively small [43-45]. The effective value of vos can often be determined by the so-called longitudinal (or "constant charge ) solvent relaxation time, rL [43, 44]. This quantity is related to the experimental Debye relaxation time, rD, obtained from dielectric loss measurements using [43]... 

Some solvents may be characterized by one Debye relaxation time, t, corresponding to the rotational diffusion in the case of alcohols and solvents, where hydrogen bonds are formed, more than one relaxation process is observed [10]. 

The change in the emission spectrum with time after pulsed excitation (TRES) is a method for assessing the overall response of the solvent to a change in solute geometry or polarity [22]. The precise values of the relaxation times depend upon the method of measurement. At room temperature the TRES solvent correlation times are subnanosecond and, in some cases subpicosecond. The Debye relaxation time in water is 8 ps, while the TRES correlation time is shorter [22]. Although there is not, in general, a... 

In a solvation process reactant molecules are always in contact with the solute and there is no operational significance to the solvent concentration [26]. The first order rate constant, in Eqs. (10)-(12), (I4)-(I7) measures the response time of the solvent to the change in the solute that follows absorption of a photon. Solvent correlation times extracted from TRES spectral shifts are a measure of the time necessary for the solvent envelope in the aggregate to adjust by rotation and translation to the excited state geometry [22]. The solvent motions required for addition of a single solvent molecule to a vacant coordination site in a five-coordinate intermediate are not necessarily the same as those that are monitored in a TRES measurement. The Debye relaxation time of the solvent, which is usually longer than the TRES correlation time of the solution, is more closely related to k. ... 

One system of particular importance is trans-Cr(NH3)2(NCS)4 in an alcohol/H20 solution at —65° where 50% of the reaction is unquenchable [46]. Although the viscosity of the solvent was not measured it was greater than 300 cP [53]. is reduced a bit more than 50% at high viscosities in polyalcohol/water mixtures of trans-Cr(NH3)2(NCS)4 at 20°C [54], Most of this decrease occurs below 3cP. The Debye relaxation time is less than lOps in water (r] = 1 cP) and 0.5ns in 1-propanol (r = 2cP). Viscosity is not an ideal measure of solvent mobility but it is unlikely that solvent motion is involved in the fast reaction when the viscosity is 300 cP. 

According to the Debye model there are three parameters associated with dielectric relaxation in a simple solvent, namely, the static permittivity s, the Debye relaxation time td, and the high-frequency permittivity Eoq. The static permittivity has already been discussed in detail in sections 4.3 and 4.4. In this section attention is especially focused on the Debye relaxation time td and the related quantity, the longitudinal relaxation time Tl. The significance of these parameters for solvents with multiple relaxation processes is considered. The high-frequency permittivity and its relationship to the optical permittivity Eop is also discussed. 

Fig. 6.2 Plot on logarithmic scales of the Debye relaxation time Xq for aprotic solvents against the product where p is the viscosity and r, the molecular solvent radius.

The dominant forces that determine deviations from ideal behaviour of transport processes in electrolytes are the relaxation and electrophoretic forces [16]. The first of these forces was discussed by Debye [6, 17]. When the equilibrium ionic distribution is perturbed by some external force in an ionic solution, electrostatic forces appear, which will tend to restore the equilibrium distribution of the ions. There is also a hydrodynamic effect. It was first discussed by Onsager [2, 3]. Different ions in a solution will respond differently to external forces, and will thus tend to have different drift velocities The hydrodynamic (friction) forces, mediated by the solvent, will tend to equalize these velocities. The electrophoretic ( hydrodynamic) correction can be evaluated by means ofNavier-Stokes equation [18, 19]. Calculating the relaxation effect requires the evaluation of the electrostatic drag of the ions by their surroundings. The time lag of this effect is known as the Debye relaxation time. 

Here T is the assumed Debye relaxation time of the solvent in ps and Kg the low frequency specific conductance of the solution in mho/m. while p and p are either 2/3 or 1 for slip or stick boundary conditions of continuum solvent at the ion surface. This prediction qualitatively explained a large fraction of the observed decrements in water and methanol (76) (79) and the relatively enormous decrements (relative to total ion concentrations) in sulfuric acid solutions (80) as a result of the large s 400 ps at 25 C (as compared to 8 ps for water and 52 ps for methanol). 

In Debye solvents, x is tire longitudinal relaxation time. The prediction tliat solvent polarization dynamics would limit intramolecular electron transfer rates was stated tlieoretically [40] and observed experimentally [41]. 

Even if we consider a single solvent, e g., water, at a single temperature, say 298K, depends on the solute and in fact on the coordinate of the solute which is under consideration, and we cannot take xF as a constant. Nevertheless, in the absence of a molecular dynamics simulation for the solute motion of interest, XF for polar solvents like water is often approximated by the Debye model. In this model, the dielectric polarization of the solvent relaxes as a single exponential with a relaxation time equal to the rotational (i.e., reorientational) relaxation time of a single molecule, which is called Tp) or the Debye time [32, 347], The Debye time may be associated with the relaxation of the transverse component of the polarization field. However the solvent fluctuations and frictional relaxation occur on a faster scale given by [348,349]... 

This approximation requires that cos. This behavior in fact follows from a Debye dielectric continuum model of the solvent when it is coupled to the solute nuclear motion [21,22] and then xs would be proportional to the longitudinal dielectric relaxation time of the solvent indeed, in the context of time dependent fluorescence (TDF), the Debye model leads to such an exponential dependence of the analogue... 

Experiments in the picosecond time range show that C(t) is non-exponential in most solvents with an average spectral relaxation time greater than the longitudinal relaxation time tl and smaller than the Debye time td-... 

So far, the discussion of concentrated electrolyte solutions has presumed that ionic relaxation is complete and so is a static correction. Dynamic electrolyte theories are still in their infancy and, in view of the rate of ionic relaxation compared with chemical reaction rates for dilute electrolytes (Sect. 1.6), such effects are probably not very important in concentrated electrolyte solutions containing reactants. The Debye— Falkenhagen [92] theory predicts a change in the relaxation time of electrolyte solutions with concentration, though experimental confirmation is scant [105]. At very high concentrations, small changes in the relaxation time ( 25%) of solvent relaxation can be identified (see also Lestrade et al. [106]). 

Here td is the so-called Debye dielectric relaxation time. One could view td as a phenomenological time constant which applies to dielectric relaxation measurements, or alternatively for simple causes, involving dielectric relaxation of weakly interacting dipoles, tD is related to the reorientation time constant of the solvent dipole in the laboratory frame. 

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... 

Solvent Relaxation time t (ps) Dipole moment (debye) Loss tangent at 2.45 GHz... 

These are obtained by introducing an explicit time dependence of the permittivity. This dependence, which is specific to each solvent is of a complex nature, cannot in general be represented through an analytic function. What we can do is to derive semiempirical formulae either by applying theoretical models based on measurements of relaxation times (such as that formulated by Debye) or by determining through experiments the behaviour of the permittivity with respect to the frequency of an external applied field. 

Utilize the results obtained in the preceding problem to calculate the relaxation times of the ionic atmospheres and the approximate minimum frequencies at which the Debye-Falkenhagen effect is to be expected. It may be assumed that Aqtjo has a constant value of 0.6. The viscosities of the solvents are as follows nitrobenzene (0.0183 poise) ethyl alcohol (0.0109) and ethylene dichloride (0.00785). 

Also, more attention should be paid to the study and analysis of electron-transfer reactions in non-Debye solvents which exhibit several relaxation times. Wider use of mixtures composed to two nonaqueous solvents of different Lewis basicity is advised. So far, such studies are rather limited (see, for instance, [227,305]). 

The plot shows a distribution closely around a slope of unity indicated by the solid line in Figure 2 except for the alcohols and nitrobenzene. Such anomaly in alcohols is also reported for other chemical processes and time-dependent fluorescence stokes shifts and is attributed to their non-Debye multiple relaxation behavior " the shorter relaxation components, which are assigned to local motions such as the OH group reorientation, contribute the friction for the barrier crossing rather than the slower main relaxation component, which corresponds to the longitudinal dielectric relaxation time, tl, when one regards the solvent as a Debye dielectric medium. If one takes account of the multiple relaxation of the alcohols, the theoretical ket (or v,i) values inaease and approach to the trend of the other solvents. (See open circles in Figure 2.)... 

A stig — dielectric increment per gm. protein per liter /r = dipole moment in debye units t H O is the relaxation time in water at 25° (correcting for the relative viscosity of water and the solvent actually employed) To = relaxation time of a sphere, of volume equal to that of the protein, in water at 25° ajb = ratio of major to minor axis, calculated from r and observed relaxation times, by the equations of Perrin (92) [Cohn and Edsall (Jd)], neglecting hydration. 

Other relationships which have been used to describe dielectric relaxation data include the Cole-Cole and Cole-Davidson equations [29]. These are preferred when a distribution of relaxation times rather than a single relaxation time is more appropriate to describe the data in a given frequency range. Nevertheless, the Debye model in its simple version or multiple relaxation versions works quite well for most of the solvents considered here. 

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