Waldens rule

Hydroxide ions also have abnormally high conductivities, and the explanation in their case is similar to that for hydrogen-ion conductivity. Protons are transferred from water molecules to hydroxide ions  

The net result is effectively the transfer of OH from one position to the next. 

An important relationship between molar conductivity and viscosity was discovered in 1906 by the Russian-German chemist Paul Walden (1863-1957). In the course of a study of the conductivity of tetramethylammoniura iodide in various solvents, Walden noticed that the product of the molar conductivity at infinite dilution and the viscosity rj of the solvent was approximately constant  

Similarly, it has been shown that the product of individual ion conductivities and viscosity is also a constant. Such relationships are most satisfactory for ions which are approximately spherical 

These relationships are not particularly surprising, since the speed with which a species diffuses through a liquid is inversely proportional to the viscosity of the liquid. Viscosity is a property which is a measure of the reciprocal of the speed with which molecules move past one another in the liquid state (see Section 11.8). 

In the continuum model, the ionic motion is considered as that of a charged sphere of radius R which carries an elementary charge through a viscous medium of viscosity T. The drag is provided by the electric force. 

A velocity gradient exists between the surface of the sphere and the bulk of the liquid which is at rest. Here it is assumed that a liquid layer adjacent to the surface sticks to the sphere (sticking condition). The sphere is attaining the constant velocity Vq and the sum of the forces on the sphere has to cancel, such that 

On the other hand, a charge carrier in a viscous medium under the influence of an externally applied electric field achieves a constant drift velocity given as 

Introducing Equation 105 in Equation 106 yields a formula for the ion mobility, 

The product of mobility and viscosity is a constant. Equation 108 is also called Walden s rule. Generally, Equation 108 should hold for different liquids as well as for a given liquid at different temperatures. For the case that a velocity gradient exists between the liquid layer in contact with the surface of the sphere itself, the factor of 6n becomes 4ti (slipping condition). This condition is fulfilled in the case of electron bubbles (see Section 7.2). Another inherent assumption in the derivation of Equation 108 is the notion that the flow of the liquid around the sphere remains laminar. 

In the beginning of the nineteenth century, Paul Walden suggests that the product of the ionic mobility and solvent viscosity is a constant value and does not depend on temperature and pressure  

Walden s rule can be obtained by simply combining Equations 3.20 and 3.21  

Conductivity (k) of 0.1 mol kg KCI(aq), Viscosity of Water q, and Their Product Divided by Density of Water p 

Therefore, using the hydrodynamic model of mobility, we can see from Equation 3.22 that the product of ionic conductivity and viscosity should theoretically be constant and independent of temperature. 

Combining Equations 3.10 and 3.22, a similar equation can be obtained for the equivalent conductivity of an electrolyte  

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If the temperature dependence of conductivity is known in a given solvent, an estimate of an unknown A0 at higher temperatures may be obtained which is much better than that measurable at lower temperatures with the help of the Walden rule ... 

It follows from Eqs. (2.6.6), (2.6.8) and (2.6.10) that the presence of the solvent has two effects on the ionic mobility the effect of changing viscosity and that of changing the ionic radius as a result of various degrees of solvation of the diffusing particles. If the effective ionic radius does not change in a number of solutions with various viscosities and if ion association does not occur, then the Walden rule is valid for these solutions ... 

Using the Walden rule, A0 was estimated and given in Table 4. Furthermore, since the limiting conductances of the triple ions would... 

Ionic liquids with discrete anions have a fixed anion structure but in the eutectic-based liquids at some composition point the Lewis or Bronsted acid will be in considerable excess and the system becomes a solution of salt in the acid. A similar scenario also exists with the incorporation of diluents or impurities and hence we need to define at what composition an ionic liquid is formed. Many ionic liquids with discrete anions are hydrophilic and the absorption of water is found sometimes to have a significant effect upon the viscosity and conductivity of the liquid [20-22], Two recent approaches to overcome this difficulty have been to classify ionic liquids in terms of their charge mobility characteristics [23] and the correlation between the molar conductivity and fluidity of the liquids [24], This latter approach is thought by some to be due to the validity of the Walden rule... 

Another equation which has received use is the Wishaw-Stokes equation, which is discussed in Robinson and Stokes [32], This equation is based on the extended Onsager equation and is corrected for viscosity in a Walden rule sort of term ... 

The Walden rule connecting viscosity and equivalent conductivity A = EXj,... 

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. 

In this case it is the silver cation that can slip through the channels set up by the alkali halide sublattice [35]. These systems can retain their high conductivity in the glassy state, as illustrated in the case of the silver-alkaU halide system in Figure 2.4b. For such systems a decoupling index has been defined from the ratio of the conductivity and fluidity relaxation times [32, 36], which can equally well be defined from the exponent a in the fractional Walden rule. 

An interesting test of the validity of the Walden rule is provided by the conductance measurements, made by LaMer and his collaborators, of various salts in a series of mixtures of light water (H2O) and heavy water (D2O). The results indicate that, although the rule holds approximately, it is by no means exact. ... 

This correction has been frequently employed when dealing with the conductance of bulky ions. Its use modifies the value of a, but for most solvents the effect is small. It is not obvious that the viscosity correction is valid since it is known that when the viscosity of a solvent is increased by addition of a nonelectrolyte, the product A ri is not constant, that is the Walden rule is not obeyed. Treiner and Fuoss increased the viscosity of acetonitrile by addition of octacyanoethyl sucrose (CES), and measured the conductance of some tetraalkylammonium salts in these media. The viscosity increased up to a hundredfold from that in pure acetonitrile. They found that the extrapolated A" does not obey the Walden rule, rather = constant for solvents with more than 40 %... 

The case is similar for the enthalpy of fusion. To a large extent, it depends on the structure of the crystal that is formed. In principle, the Clausius-Clapeyron equation can be applied however, the information about the pressure dependence of the melting point is usually not available. The so-called Walden rule [13] gives at least an estimation for aromatic compounds like benzene or naphthalene ... 

A nonstoichiometric mixture was produced from a poly-oxometalate and a heteropolyadd to make a proton conductor with the general formula of H3 xM POM. On the Walden plot, this mixture is above the line predicted by the classical Walden rule and in the superionic liquid region, indicating conduction is occnning by a more efficient conduction mechanism than the Walden mechanism, possibly by the Grotthus mechanism. ... 

Schreiner C, Zugmann S, Hartl R, Gores HJ (2010) Fractional Walden rule for ionic liquids examples from recent measurements and a critique of the so-called Ideal KQ line for the Walden plot. J Chem Eng Data 55 1784-1788... 

The Walden Rule states that the product of the hmiting molar conductivity and the pure solvent s viscosity rj constant for infinitely diluted electrolyte solutions. It is a nice approximation for dilute solutions but shows rather large deviations for pure salts. The modified Walden mle, see Eq. la, introduces an additional empirical parameter, a, to cope with this deviation, see Eq. lb, Ic. 

The molar conductivity Am of the liquid salt is obtained from the specific conductivity k and the molar volume Vm (Eq. Id) of the salt that is available from density measurements. The modified Walden rule is applied for comparing ILs. The deviation from the Walden rule represented by a is interpreted in terms of ionic-ity. A small ionicity is caused by strong ion-ion interaction. 

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