Stokes-Einstein equation ionicity

The Stokes-Einstein equation connecting diffiisivity D, of ionic species i of charge z, and radius r with the viscosity q of the medium in which the diffusion is occurring. 

The Walden rule is interpreted in the same manner as the Stokes-Einstein relation. In each case it is supposed that the force impeding the motion of ions in the liquid is a viscous force due to the solvent through which the ions move. It is most appropriate for the case of large ions moving in a solvent of small molecules. However, we will see here that just as the Stokes-Einstein equation applies rather well to most pure nonviscous liquids [30], so does the Walden rule apply, rather well, to pure ionic liquids [15]. When the units for fluidity are chosen to be reciprocal poise and those for equivalent conductivity are Smol cm, this plot has the particularly simple form shown in Figure 2.6. 

All transport processes (viscous flow, diffusion, conduction of electricity) involve ionic movements and ionic drift in a preferred direction they must therefore be interrelated. A relationship between the phenomena of diffusion and viscosity is contained in the Stokes-Einstein equation (4.179). 

This relationship is known as the Stokes-Einstein equation. Strictly speaking it should only be applied at infinite dilution to monoatomic ions. However, in practice it is applied to more complex ions and at finite ionic strengths. If the diffusion coefficient for the ion is measured experimentally, an effective radius for the ion can be estimated using the viscosity of the pure solvent. 

By combining the Stokes-Einstein equation (equation (6.7.27)) with equation (6.7.23), the following expression for the limiting ionic molar conductance is obtained ... 

For a given water content and by applying the Stokes-Einstein equation and the rate process to ionic migration s , it is possible to calculate, for two dilferent ions, the difference between their activation free energies for the elementary ion transfer reactions, the proton being chosen as the reference ion... 

In a simple hydrodynamic approach, transport parameters such as the ionic conductivity <7 , the diffusion coefficient D , and the electrochemical mobility Ui of ionic/atomic species i in Uquid/glassy systems are linked to the dynamic viscosity t] by the Stokes-Einstein equation ... 

For all known systems, the material-dependent parameter B, in Eq. (12.14) does not necessarily have the same value as the parameter B, in Eq. (12.18a). In a system with a higher probability of ionic charge motions than for viscous flow events, the strict coupling between dynamic viscosity and conductivity via the Stokes-Einstein equation does not hold [33, 34]. By introducing another independent material parameter, B B, Eq. (12.18a) can be rewritten as... 

Figure 1.4 Comparison of the acmal case of diffusion in an ionic liquid and the model from which the Stokes-Einstein equation is derived.

This work, along with all other diffusion studies in ionic liquids, depends on the validity of the Stokes-Einstein equation and here lies the main discrepancy with the analysis of most difhision coefficient data in ionic liquids. To be strictly accurate, the distance term, R, in Eqnation 1.3 should be replaced by the correlation length, which is only really equivalent to the radius of the diffusing species when the size of the diffusing particle is large conpared with the solvent particles [27]. Eqnation 1.3 was initially derived to describe the random movement of... 

Classical diffusion can be described by Equation 1.3 when the radius of the sphere is small conpared with the mean free path. With ionic liquids, the mean free path can be less than the radius of the ion, and hence the ion can be considered as moving via a series of discrete jumps where the correlation length is a measure of the size of the hole into which the ion can junp. Appreciating why deviations from the Stokes-Einstein equation occur shows why a model based on holes becomes appropriate. The approximate nature of the Stokes-Einstein equation is often overlooked and is discussed in detail by Bockris and Reddy [5, p. 379]. There are numerous aspects that need to be taken into account, including that it is derived for non-charged particles, it is the local viscosity rather than the bulk that is required, and the ordering effect of the ions exhibits an additional frictional force that needs to be explained. 

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). 

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 ... 

In this equation -q is the viscosity, sometimes called the dynamic viscosity, of the medium, and n is the radius of the particle. Equating these two equations of ionic mobility gives the Stokes-Einstein equation [Eq. (D.18)] ... 

New expressions of ionic mobility can be obtained [Eqs. (D.23) to (D.25)] by combining Stokes-Einstein equation with Einstein equation ... 

The Stokes-Einstein equation [77] is generally applicable for molecular liquids and ILs, at least semi-quantitatively, for analyzing the diffusion coefficient D. Water is ubiquitous in the environment. It was shown that water induced accelerated ion diffusion compared to its effect on neutral species in ionic liquids [78]. We also observed that IL conductivity is significantly increased when [Bmim][BF4] is... 

The conductivity of ionic liquids can be modeled in the same manner as the viscosity, i.e. despite the high ionic strength of the liquid, ionic migration is limited by the availability of suitably sized voids [130]. Since the fraction of suitably sized holes in ambient temperature ionic liquids is effectively at infinite dilution, migration should be described by a combination of the Stokes-Einstein and Nernst-Einstein equations. This is explained in greater detail in Chapter 11.3 on process scale-up but it is sufficient to say that an expression can be derived for the conductivity, k... 

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. ... 

Harris KR (2010) Relations between the fractional stokes- einstein and nemst- einstein equations and velocity correlation coefficients in ionic liquids and molten salts. J Phys ChemB 114 9572-9577... 

Results of the EPR study of rotational diffusion (Tc) in viscous ionic hquids BmimBF4 and BmimPFe with proxyl radical (R5COOH) in a molecular and anionic forms were analysed in (Miyake et al., 2009). The experimental average values and activation energies for rotational diffusion disagreed with those calculated from fractional Stokes-Einstein-Debye equations. 

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