Diffusion transference number determination

The classical methods of experimental transference number determination can be divided into three general groups. The first (the Hittorf method) is essentially an analytical approach, which relates changes in cell composition to the transference numbers of the electrolyte solution. The second group of methods relates the motion of the boundary separating zones of different composition to the transference numbers. The final approach relates the cell potential, which arises from the diffusion potential, to the transference number. Each of these methods is summarised, in turn, below. 

Distinct from the preceding methods is a crude approach to transference number determination. This method relies on the relationship between the mobilities and diffusion coefficients of ions, the Einstein law ... 

If the ions diffusion coefficients are measured independently, the ratio of mobilities and transference numbers can be found (equation (20.1.2-9)). The problem is that the Einstein law is only valid at infinite dilution. This approach to transference number determination is substantially more limited than the schemes detailed above. 

The overall permeation rate of a material is determined by both ambipolar conductivity in the bulk and interfacial exchange kinetics. For -> solid electrolytes where the electron - transference numbers are low (see -> electrolytic domain), the ambipolar diffusion and permeability are often limited by electronic transport. 

The -> concentration cells are used only for determination of -> transport (transference) numbers, - activity, and -> activity coefficients of electrolytes and other quantities. Their practical application is limited by the -> selfdischarge due to the spontaneous diffusion process. In concentration cells no chemical reactions occur, a physical process (the equalization of activities by diffusion) causes the potential difference and supplies the energy. 

The electrical conductance, transference number, and diffusion coefficient provide the three relations from which the phenomenological coefficients can be determined, and for a monomonovalent salt we have... 

Hittorf Method Experimental Procedure.—In Hittorf s original determination of transference numbers short, wide electrolysis tubes were used in order to reduce the electrical resistance, and porous partitions were inserted to prevent mixing by diffusion and convection. These partitions are liable to affect the results and so their use has been avoided in recent work, and other precautions have been taken to minimize errors due to mixing. Many types of apparatus have been devised for the determination of transference numbers by the Hittorf method. One form, which was favored by earlier investigators and is still widely used for ordinary laboratory purposes, consists of an H-shaped tube, as shown... 

Although the Hittorf method is simple in principle, accurate results are difficult to obtain it is almost impossible to avoid a certain amount of mixing as the result of diffusion, convection and vibration. Further, the concentration changes are relatively small and any attempt to increase them, by prolonged electrolysis or large currents, results in an enhancement of the sources of error just mentioned. In recent years, therefore, the Hittorf method for the determination of transference numbers has been largely displaced by the moving boundary method, to be described later. 

Figure 13-5 summarizes data for in terms of Pe,.. In terms of the radial Peclet number determined from mass-transfer data, if dp is the particle diameter and u is the superficial velocity, the turbulent-diffusion contribution is... 

Transport parameters, which appear in the various forms of the infinite dilute transport equations, are the electrolyte conductivity, the ion mobility, the ion diffusion coefficient, and the ion transference number. All of these parameters can be determined from ionic equivalent conductances with units of (5-cm )/equiv) of cations and anions in solution. The ion mobility M , which appears in Equation (26.54), is related to by... 

Figure 22.32a. The accumulation of chloride ions in the occluded solution reaches a stationary level, at which the rate of inward ion migration equals the rate of outward diffusion of the ions. Furthermore, the anodic chloride ion migration is accompanied by an electroosmotic flow of water molecules into the occluded solution. The final chloride concentration thus established in the steady state is determined by the ratio of the chloride ion migration rate to the electroosmotic water flow rate. It is therefore the transference numbers, Ter an(J tH2o, of chloride ions and water molecules that determine the chloride ion concentration in the occluded solution.

Consequently the mass transfer rate during diffusion combustion of polymers is determined by the ratio of the heat of combustion to the effective enthalpy of polymer gasification. The lower the combustion heat and the higher the polymer gasification enthalpy or, in other words the more heat resistant the polymer, the lower is the B value. For polymer combustion in air the B value of e.g., PMMA varies between 1.3-1.4, that of polyethylene between 0.5-0.6, of phenolic resins between 0.14-0.4 74 75). An increase of the oxygen concentration in the oxidative medium and of the oxidant temperature causes a rise of the mass transfer number B. Lower B values have been observed in thermally stable polymers of the carbonizable types. 

The broadening of the MARY lines was also analyzed to study the degenerate electron exchange in the irradiated alkane solutions involving cw-decalin radical cation [21] and hexafluorobenzene radical anion [20, 21]. It was found that for cis-decalin radical cation, the rate of charge transfer was determined by the number of diffusion collisions, while for hexafluorobenzene radical anion, it was lower and reached the diffusion-controlled limit only at high viscosity. 

For low wind speeds, one might ask, at what characteristic depth or distance from the interface is the rate of advective transfer of normal dissolved gases away from the interface equal to diffusive transfer To answer this question, we can use the dimensionless Peclet number, (dV/D), which expresses the relative importance of mass transfer by advection to transfer by diffusion. In the Peclet number, d can be taken as the thickness of the diffusive layer, V the velocity and D as the gas diffusivity in the water phase. If we take V as the piston velocity with an appropriately low value of about 1 cm h, and a typical diffusivity for gases in water of about 10" cm s the thickness of the boundary layer can be determined for a Peclet number, Pe= 1, i.e. at a distance from the interface where advective and diffusive transport are comparable. Under these conditions, d is... 

Several ingenious methods of measuring transference numbers will be described because even today one cannot buy off-the-shelf transference kits suitable for research. Some of these methods have been developed and adapted to make them suitable for determinations under extreme conditions of concentration, temperature and pressure while others have remained historical curiosities. The absolute values of transference numbers and their variations with concentration have provided essential insight into the structure of ionic solutions. The triad of conductance, transference number and diffusion coefficient now furnishes a valuable basis for understanding the flow properties of electrolytes. 

Transference numbers have formed one of the cornerstones in our understanding of electrolyte solutions. Hittorf s discovery in 1853 that transference numbers depended on the ion, the co-ion, and the solvent proved that each ion in a given solvent possesses its own individual mobility. Even today ionic mobilities must be determined by a combination of transference and conductance experiments for we still cannot predict their values accurately from first principles The importance of ionic mobilities can hardly be overemphasized since they are the only properties of individual ions that can be unambiguously measured (either directly or via trace diffusion coefficients). They therefore provide unique insight into ion-solvent interactions. Hittorf s later transference experiments also revealed the existence and composition of a variety of complex ions in solution. His approach has been followed in more recent structural investigations, for instance in studying the complex ions present in aluminium plating solutions (7A,). 

Because the specific conductivity k (S/m) of an electrolyte is determined readily and easy, this property is widely used for optimizing the battery performance. In contrast, other parameters which are more difficult to obtain, e.g., diffusion coefficients of ions near to or in the electrode materials or transference numbers of ions, are seldom studied and not yet included in optimization. We expect that automated measurement systems will be used in the future to optimize this and other critical parameters of solutions as long as no valid theoretical approach is available. These systems should be able to measure selected quantities automatically as a function of temperature and composition of solutions according to proposals made by optimization methods such as simplex. First steps on this way were undertaken by Schweiger et al., who presented an equipment that is able to measure K(T(t)) and T(t) automatically in up to 32 cells [34-38]. 

Applying a constant current for a known time leads to a concentration gradient that is measured indirectly by observing the relaxation potential after current interruption [16]. Combining the galvanostatic polarization experiment, determination of salt diffusion coefficient and concentration dependence of the potential by emf measurements enables to calculate the cationic transference number [17]. 

The molar conductivity of an electrolyte is the more generally useful quantity since the Kohlrausch law allows its limiting value to be resolved into those of its constituent ions. Comparison between different electrolytes with a common ion therefore allows the determination of an unknown molar conductivity. However, the quantity typically measured is the overall electrolytic conductivity. A way to apportion the conductivity (and hence mobility) to the individual ions of the electrolyte is required. Equation (20.1.2-11) shows that resolution of the molar conductivity into the terms arising from its constituent ions is possible if the transference number of the ion is found. Although this property and the methods developed to measure it may seem rather arcane, it has been of fundamental importance in the understanding of the conductivity and diffusion potentials developed within electrolyte solutions. Experimentally, a number of ways of measuring transference numbers have been developed these are summarised below. 

Finally, a simple measurement of a diffusion potential allows us to determine the potassium ion transference number as shown in the cell of Figure 20.7. 

Ionic Conductivities in Aqueous Solutions The thermodynamic quantities for ions in solution dealt with in the previous sections could be measured only for complete electrolytes (or for charge balanced differences between ions of the same sign) but not for individual ions. On the contrary, this is not the case for ionic conductivities (and diffusion coefficients, see Section 2.3.2.2). These can be determined experimentally for individual ions from the electrolyte conductivities and the transference numbers. The conductivity of an electrolyte solution is accurately measured with an alternating external electric field at a rate of lkHz imposed on the solution with a high impedance instrument in a virtually open circuit (zero current). The molar conductivity, Ag, can then be determined per unit concentration. Ion-ion interactions cause the conductivities of electrolytes to diminish as the concentration... 

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