Stokes-Einstein product

The validity of Eq. (12) for correlations of limiting-current measurements was first questioned by Arvia et al. (A5), and later by Wragg and Ross (W13a). The latter found that limiting currents in an annular flow cell could be correlated in better agreement with the Leveque mass-transfer theory if a lower mobility (Stokes-Einstein) product were employed, such as... 

Table 21.2.6. Diffusion coefficients and Stokes-Einstein products of a number of species studied in the 40-60 moi% [Cj-mimjCi-AiCis ionic iiquids at 40°C...

This is of course not the case when working with room temperature ionic liquid systems. Electrochemical and spectroscopic studies of cobalt, copper, and nickel, have been carried out in the AlClj-butylpyridinium chloride molten salt system. The direct current and pulsed current electrodeposition of Ni-Al alloys has also been shown in acidic AlCls-butylpyridinium chloride ionic liquids. This particular alloy has also been shown to be successful in AlCl3-[C2-mim]Cl ashave Co-Al andCu-Al. Electrochemical techniques can also be used to calculate the diffusion coefficients of metal ions. Table 21.2.6 shows the calculated diffusion coefficients and stokes-Einstein products of cobalt(II), copper(I), nickel(II) and zinc(II) in the 40-60 mol% [Cj-mimlCl-AlClj ionic liquid. 

The Stokes-Einstein equation predicts that DfxITa is independent of the solvent however, for real solutions, it has long been known that the product of limiting interdiffusion coefficient for solutes and the solvent viscosity decreases with increasing solute molar volume [401]. Based upon a large number of experimental results, Wilke and Chang [437] proposed a semiempirical equation,... 

Walden s rule — This empirical rule states that the product of the equivalent -> conductivity and the viscosity of the solvent for a particular electrolyte at a given temperature is a constant. Its rational explanation is based on the Stokes-Einstein equation that connects the... 

In experiments involving radiotracer measurements of diffusion in molten salt, the Stokes-Einstein equation has been found to be roughly applicable. For a series of ions, in molten salts it was found that the product D /T = 10 dyn mol . From this information, find whether the best form of the coefficient in this expression for this case is nearer to 6 or 4. 

This equation was deduced in Section 4.4.8. It is of interest to inquire here about its degree of appiicabiiity to ionic liquids, i.e., fused salts. To make a test, the experimental values of the self-diffusion coefficient D and the viscosity tj are used in conjunction with the known crystal radii of the ions. The product D r//T has been tabulated in Table 5.22, and the plot of D tj/T versus 1/r is presented in Fig. 5.31, where the line of slope k/6n corresponds to exact agreement with the Stokes-Einstein relation. ... 

Thus, the combined SE and the DSE equations predict that the product Dtxc = (A Tc)sedse should equal 2r /9. Measurements of probe translational diffusion and rotational diffusion made in glass-formers have found that the product Dtr can be much larger than this value, revealing a breakdown of the Stokes-Einstein (SE) relation and the Debye-Stokes-Einstein (DSE) relation. There is an enhancement of probe translational diffusion in comparison with rotational diffusion. The time dependence of the probe rotational time correlation functions tit) is well-described by the KWW function,... 

The former is directly proportional to the square of the droplet size according to the Stokes-Einstein expression. In spite of the broad droplet size distribution and the changing droplet size with time, measuring the creaming rate can still provide a useful method for determining the relative droplet sizes of different emulsion products. The ultimate emulsion stability is not solely controlled by... 

For instance, dielectric relaxation times usually are close to, and scale simply with, the shear and bulk viscosity relaxation times, and t . Since the Stokes-Einstein equation connecting diffusivity with shear viscosity is generally valid, this implies that the structural relaxation time for a system could be estimated if the diffusivity were known. Indeed for a number of simple polar molecules the product seems experimentally to be roughly constant at 2X 10 cm, implying that perturbations of a liquid of diffusivity 1 X 10 cm /sec would require of the order 10 psec to be relaxed to 1/eth of their initial value. 

The Stokes-Einstein equation for liquid-phase ordinary molecular diffusion coefficients in binary mixtures suggests that the product of Hab and the solvent viscosity /u-b should scale linearly with temperature T. Cite references (i.e., equations) from the literature and evaluate the product of Hab and /xb in terms of its scaling-law dependence on temperature for low-density gases. In other words ... 

In almost all of the work reported so far on electrochemical reactions in supercritical fluids, the emphasis has not been on preparative organic synthesis. One of the principal questions tackled has been whether the Stokes-Einstein relationship between diffusion and viscosity continues to be valid for an SCF, and the conclusion is that it probably is. In synthesis, the only significant study is the production of dimethyl carbonate from CO and methanol using sc CO2 as a cosolvent with methanol (Dombro et al., 1988). The use of sc CO2 in place of water enabled the reaction to be carried out near the critical point, that is, at a lower temperature. Tetrabutylammonium bromide (TBAB) in a concentration of 1-5% was used as an electrolyte and bromide source. The reactions occurring at the two electrodes are... 

Anisotropic diffusivity can be accommodated if the scalar product is not used. However, this extension is usually invahd. Frej and Prieve (1993) measured R as a fimction of time for a single particle near a sohd surface using an optical technique and calculated as a function of h. The theory behind Equation 8.75 is such that it can be extended to Z , in which direction the system is homogeneous, but not D. Nevertheless, such an extension was used to calculate from the experimental data, and these values agreed well with those calculated using hydrodynamics, in Equation 8.54. Their measurements show the smallest measured value to be as low as 0.015 of that in the bulk (Stokes-Einstein) at about 100 run separation distance. 

Hereby, B, A and Tq are material-dependent parameters. The parameter is proportional to the activation energy of ionic transport. In a system with a strict coupling between dynamic viscosity and conductivity, as described by the Stokes-Einstein equation, the parameter B in (8.8) is equal to the parameter B in (8.10). In a system with a higher probability for the motion of ionic charge carriers than for viscous flow events, as it can be found in case of cooperative proton transport mechanisms, the strict coupling between dynamic viscosity and conductivity does not hold [56-58]. In this case the parameter Bg in (8.10) will be smaller than B in (8.8). Combining (8.8) and (8.10) and considering the concentration dependence of cr, by introduction of the molar conductivity one will yield a fractional Walden rule (-product) as shown in (8.11). 

The average duration of an encounter (/enc) may now be deduced. It is the product of the number of recollisions and the average time between them, i.e., iVenc recoii- Inserting the expressions above, one obtains /enc ab/ ab- Since Dab oc l/fAB according to the Stokes-Einstein relation (if ta = rn), this result implies that /enc should vary approximately as r g. The calculated values [23,c] are in accordance with this expectation, and are in the region of 5 to 200 ps. 

Figure 3.7 Deviations of diffusion coefficients from the Stokes-Einstein value (2). Comparison of the diffusion coefficients of symmetrical solutes in five protic and aprotic solvents with the prediction of the Stokes-Einstein equation. The diffusion coefficient — viscosity product Drj is plotted against the reciprocal of the solute radius the predictions of Stokes law for perfect stick (kT/6n) and perfect slip kT/An) are shown by dotted lines. See text. From Ref. [13,b].

Although many out of the hundreds of reactions in aqueous solution studied by pulse radiolysis are due to the secondary products such as OH-, there are also several hundred in which an oxidising agent is reduced by the solvated electron itself, before it has had time to react with the solvent, e.g., by addition e q -I- A" A" (cf. Section 1.2.3). In these reactions the solvated electron may be regarded as a distinct ionic entity, with a definite diffusion coefficient determined from the conductivity), and a radius r of about 3 A (calculated from the Stokes-Einstein equation. Section 1.2.3). The electron lies in a potential-energy trap ... 

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