Ionic atmosphere relaxation

Another arena for the application of stochastic frictional approaches is the influence of ionic atmosphere relaxation on the rates of reactions in electrolyte solutions [19], To gain perspective on this, we first recall the early and often quoted triumph of TST for the prediction of salt effects, in connection with Debye-Hiickel theory, for reaction rates In kTST varies linearly with the square root of the solution ionic strength I, with a sign depending on whether the charge distribution of the transition state is stabilized or destabilized by the ionic atmosphere compared to the reactants. 

In Section 12.1 it was shown that when the central reference ion moves under the influence of an external field there is an excess of negative charge behind the cation and a deficit of negative charge in front of it. If the external field is removed, the asymmetric ionic atmosphere relaxes back to the symmetric distribution. This process can be pictured as if some negative charge... 

The work presented in this chapter was performed in moderately polar media [primarily tetrahydrofuran (THF)], in which electrostatic interactions between ions are considerably larger than the thermal energy, kT. Therefore, all results are interpreted in terms of specific ion-pairing effects rather than the bulk ionic atmosphere relaxation. 

When an external electric field is imposed on an electrolyte solution by electrodes dipped into the solution, the electric current produced is proportional to the potential difference between the electrodes. The proportionality coefficient is the resistance of the solution, and its reciprocal, the conductivity, is readily measured accurately with an alternating potential at a rate of 1 kHz in a virtually open circuit (zero current), in order to avoid electrolysis at the electrodes. The conductivity depends on the concentration of the ions, the carriers of the current, and can be determined per unit concentration as the molar conductivity Ae. At finite concentrations ion-ion interactions cause the conductivities of electrolytes to decrease, not only if ion pairs are formed (see Sect. 2.6.2) but also due to indirect causes. The molar conductivity Ae can be extrapolated to infinite dilution to yield Ae" by an appropriate theoretical expression. The modern theory, e.g., that of Fernandez-Prini (1969), takes into account the electrophoretic and ionic atmosphere relaxation effects. The molar conductivity of a completely dissociated electrolyte is ... 

The most characteristic properties of ions are their abilities to move in solution in the direction of an electrical field gradient imposed externally. The conductivity of an electrolyte solution is readily measured accurately with a 1 kHz alternating potential in a virtually open circuit, in order to avoid electrolysis. The molar conductance of a completely dissociated electrolyte is A2 = A2°° - 2 + EC2 In C2 + J iR ) C2 — J" R")c2, where S, E, f, and f are explicit expressions, containing contributions from ionic atmosphere relaxation and electrophoretic effects, the latter two depending also on ion-distance parameters R. The infinite dilution can be split into the limiting molar ionic conductivities by using experimentally measured transport numbers extrapolated to infinite dilution, t+° and i °° = 1 - <+°°. For a binary electrolyte, Aa = 2+°° -I- and = i+ A2. Values of the limiting ionic molar conductivities in water at 298.15 K [1] are accurate to 0.01 S cm mol (S = Q ). 

Ideas concerning the ionic atmosphere can be used for a theoretical interpretation of these phenomena. There are at least two effects associated with the ionic atmosphere, the electrophoretic effect and the relaxation effect, both lowering the ionic mobilities. Formally, this can be written as... 

The relaxation effect arises because a certain time, is required for the formation or collapse of an ionic atmosphere around the central ion. When an ion moves in an electric held, its ionic atmosphere lags somewhat behind, as it were its center (Fig. 7.7, point B) is at a point where the central ion had been a little earlier. The conhgurahon of the ionic atmosphere around the central ion (point A) will no longer be spherical but elongated (ovoid). Because of this displacement of the charges, the ionic atmosphere has an electrostahc effect on the central ion which acts in a direction opposite to the ion s motion. A rigorous calculation of this effect was made in 1927 by Lars Onsager. His solution was... 

Fig. 2.7 Time-of-relaxation effect. During the movement of the ion the ionic atmosphere is renewed in a finite time so that the position of the ion does not coincide with the centre of the ionic atmosphere... 

Debye and Falkenhagen predicted that the ionic atmosphere would not be able to adopt an asymmetric configuration corresponding to a moving central ion if the ion were oscillating in response to an applied electrical field and if the frequency of the applied field were comparable to the reciprocal of the relaxation time of the ionic atmosphere. This was found to be the case at frequencies over 5 MHz where the molar conductivity approaches a value somewhat higher than A0. This increase of conductivity is caused by the disappearance of the time-of-relaxation effect, while the electrophoretic effect remains in full force. 

Debye relaxation time phys chem According to the Debye-Hrickel theory, the time required for the ionic atmosphere of a charge to reach equilibrium in a current-carrying electrolyte, during which time the motion of the charge is retarded. da bT, re,lak sa-sh3n, tTm ... 

This effect is called the relaxation effect. Second, in the presence of the ionic atmosphere, a viscous drag is enhanced than in its absence because the atmosphere moves in an opposite direction to the moving ion. This retarding effect is called the electrophoretic effect. In Eq. (7.1), the Ah°°-term corresponds to the relaxation effect, while the E-term corresponds to the electrophoretic effect. For details, see textbooks of physical chemistry or electrochemistry. 

The concentration dependence of ionic mobility at high ion concentrations and also in the melt is still an unsolved problem. A mode coupling theory of ionic mobility has recently been derived which is applicable only to low concentrations [18]. In this latter theory, the solvent was replaced by a dielectric continuum and only the ions were explicitly considered. It was shown that one can describe ion atmosphere relaxation in terms of charge density relaxation and the elctrophoretic effect in terms of charge current density relaxation. This theory could explain not only the concentration dependence of ionic conductivity but also the frequency dependence of conductivity, such as the well-known Debye-Falkenhagen effect [18]. However, because the theory does not treat the solvent molecules explicitly, the detailed coupling between the ion and solvent molecules have not been taken into account. The limitation of this approach is most evident in the calculation of the viscosity. The MCT theory is found to be valid only to very low values of the concentration. 

The problem of making significant dielectric constant measurements in these ranges is to separate the relaxation effects of the ionic atmosphere around the ions (Chapter 3) from effects connected with ion-solvent interactions. At low concentrations (< 0.01 mol dm" ), the former effects are less important, but at such concentrations the decrements in the dielectric constant are too small for accurate measurement. Theoretical work makes it clear that a series of measurements over a large range of frequencies (e.g., 1 Hz to 1 GHz) are needed to separate dielectric effects from those due to relaxation of the ionic atmosphere. 

The actual mechanism by which the ions constituting the ionic atmosphere are dispersed is none other than the random-walk process described in Section 4.2. Hence, the time taken by the ionic cloud to relax or disperse may be estimated by the use of the Einstein-Smoluchowski relation (Section 4.2.6)... 

What distance jc is to be used hi other words, when can the ionic cloud be declared to have dispersed or relaxed These questions may be answered by recalling the description of the ionic atmosphere where it was stated that the charge density in a dr-thick spherical shell in the cloud declines rapidly at distances greater than the... 

One kind of relaxation time has already been discussed in Section 4.6.8, namely, the time of adjustment to dissymmetry of the ionic atmosphere around an ion when an applied electric field is switched on. Its understanding is basic to our picture of ionic... 

Eventually there is a critical frequency above lO s at which the ionic cloud cannot adjust anymore to the ion s movements in the right way because there is too much inertia to execute the rapid changes required by the oscillating applied field. The reciprocal of this critical frequency is called the relaxation time of the asymmetry of the ionic cloud. As a consequence, an increase in conductivity occurs at this frequency because there is no longer more charge behind the ion than in front. This increase in conductance at the critical frequency is called the Debye effect. It is part of the evidence that shows that the ionic atmosphere is indeed present and functioning according to the way first calculated by Debye. 

These ideas about relaxation times are applied to several phenomena, including the changes in asymmetry of the ionic atmosphere and the unusual behavior of the dielectric constant of water, which has three values according to the frequency with which it is measured. They are 78, 5, and 2. 

Utilize the calculated values of the thickness of the ionic atmosphere R in 0.1 N solutions of a univalent electrolyte in (a) nitrobenzene, (b) ethyl alcohol, and (c) ethylene dichloride to calculate the relaxation times of the ionic atmospheres. (Constantinescu)... 

In the text, a discussion of what happens to an ionic atmosphere when the ion at its center is discharged gives rise to an equation for the relaxation time of the ion atmosphere (as it disperses). Find such an expression. Apply it to find the time the ionic atmosphere takes to relax around Na ions in a 0.01 M NaCl solution when the diffusion coefficient of Na" " is 1.93 x 10 ... 

Ohm s law implies that the equivalent conductivity is independent of the strength of the applied electric field. This is certainly so for a very wide variety of applied fields, 1 to 10 V cm, in fact. Howevo, Wien showed that (with appropriate precaution taken against heating of the solution, etc.), the equivalent conductivity of electrolytes undergoes a substantial increase at about 10 V cm . By appropriate consideration of the ionic atmosphere and its time of relaxation, show that a credible model to explain the above is that the high applied field... 

Conductance with High Potential Gradients.—When the applied potential is of the order of 20,000 volts per cm., an ion will move at a speed of about 1 meter per sec., and so it will travel several times the thickness of the effective ionic atmosphere in the time of relaxation. 

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). 

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