Transference number variation with concentration

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. 

Figure 7. The variation with concentration of the transference numbers of several 1 1 electrolytes in water at 25°C (Reproduced from Ref. 56. Copyright 1932 American Chemical Society.)...

Fig. 42. Variation of transference number with concentration of indicator solution...

Influence of Temperature on Transference Numbers.—The extent of the variation of transference numbers with temperature will be evident from the data for the cations of a number of chlorides at a concentration of 0.01 N recorded in Table XXX these figures were obtained by the Hittorf method and, although they may be less accurate than those in Table XXIX, they are consistent among themselves. The transference... 

It will be observed from the results in Table XXIX that transference numbers generally vary with the concentration of the electrolyte, and the following relationship was proposed to represent this variation, viz.,... 

Since the Onsager equation is, strictly speaking, a limiting equation, it is more justifiable to see if the variation of transference number with concentration approaches the theoretical behavior with increasing dilution. The equivalent conductance of a univalent ion can be expressed in the form of equation (37), page 90, viz.. 

The first two terms on the right-hand side of equation (28) may be evaluated directly from the experimental data, after deciding on the concentration Co which is to represent the ref( rence state. The third term is obtained by graphical integration of 8 against Ey the value of 5 being derived from the known variation of the transference number with concentration. 

The use of equation (30) gives a mean transference number of the electrolyte within the range of concentrations from Ci to C2, but this is of little value because of the variation of transference numbers with concentration a modified treatment, to give the results at a series of definite concentrations, may, however, be employed. If the concentrations of the solutions are c and c + dc, the e.m.f. of the cell with transference is given by the general form of equation (22) as... 

As explained before, the open-circuit potential of the battery depends on concentration, temperature, and transport limitations. The real voltage delivered by a battery in a closed circuit is affected by ohmic limitations (ohmic potential), concentration limitations (concentration overpotential), and surface limitations (surface overpotential). The close circuit potential of the cell is given by the open-circuit potential of the cell minus the drop in potential due to ohmic potential, concentration overpotential, and surface overpotential. The ohmic potential is due to the ohmic potential drop in the solution. It is mostly affected by the applied charge/discharge current of the battery. The concentration overpotential is associated with the concentration variations in the solution near the electrodes. It is strongly affected by transport properties such as electrolyte conductivity, transference number, and diffusion coefficients. Finally, the surface overpotential is due to the limited rates of the electrode reactions. 

This equation contains only measurable quantities. However in order to integrate it, information concerning the point to point variation of the concentrations in the boundary is necessary, since the values of the transference numbers and the activity coefficients depend both upon the total concentrations of the solutions I and II and upon the proportions in which these solutions are mixed. The distribution of electrolytes in the boundary assumed by Planck and by Henderson have already been discussed. These were chosen, it is well to repeat, not because of their inherent probability, but because with them analytical integrations could be carried out. 

The transference numbers of certain electrolytes, however, show variations from the predictions of the theory. The data for silver nitrate are plotted in Fig. 5 and it is seen that the variation of the transference number of that substance with concentration is in the opposite direction from that required by the theory. A somewhat similar behavior is observed with the data on potassium nitrate, the observed values changing much more rapidly with increasing concentration than would be expected from equation (28). However, as will be seen from a study of the conductance measurements, nitrates are somewhat abnormal. 

The variation of the cation transference number with the concentration for electrolytes of higher valence type is given by the equation15... 

The results of ion transference number measurement summarized in Table 12.3, shows the variation of ionic and electronic conductance of the NC s with increasing CNP concentration. The results show an increase in electronic conductivity with CNP concentration. 

Equation 2.18 effectively incorporates the retardation effects into the mobility determination for high concentration solutions. As an example, for aqueous solution at room temperature T = 298K), using D = 78.56 and t] = 0.008948, the variation of the mobility of the positive ion with concentration in 1,1 valency electrolytes of HCl, KNO3, and NaCl are plotted in Figure 2.5 according to Equation 2.18. The variation of the transference numbers of the cations with the concentration are also plotted to discern its effect on the mobility of each ion. As observed, the square root model represents the reduction of the mobility of each ion with increasing concentration, where the reduction appear to be mostly dependent on A. ... 

For reasons given later the discussion in this section will be written in terms of symmetrical and completely dissociated electrolytes. In the event of ion association the transference number is not affected directly since always equals and these cancel out in eqn. 5.8.3. There remains the indirect effect, in that the concentration c in the interionic terms must be replaced by the ionic concentration cue. The variation of the transference number with concentration is then even smaller than is... 

Test of Eqn. 5.10.14 for the Variation of Transference Numbers with Concentration at 25°C... 

The solid solution KCl-RbCl differs basically from the solid solution NiO-MgO in two ways. Firstly, the system KCl-RbCl exhibits purely ionic conduction. The transport numbers of electronic charge carriers are negligibly small. Secondly, a finite transport of anions occurs. Because of these facts, the atomic mechanism of the solid state reaction between KCl and RbCl is essentially of a different sort than that between NiO and MgO. Once again, the diffusion profile exhibits an asymmetry (see Fig. 6-1). However, in this case the asymmetry arises not so much because of the variation of the defect concentration with composition, but rather because of the different mobilities of the ions at given concentration. Were the transport number of the chloride ions negligible, then the diffusion potential (which would be set up because of the different diffusion velocities of potassium and rubidium) would ensure that the motion of the two cations is coupled. If, on the contrary, the transference number of the chloride ions is one, then there is no diffusion potential, and the motion of the two cations is decoupled. 

Side reactions can introduce error into the measurement of physical properties in three ways [73]. Current is consumed by the side reaction, introducing error into calculations of the amount of current that went into the main reaction. Bulk concentrations of salt or solvent may change if the side reaction is substantial, and soluble products of reaction may affect the activity of the electrolyte. Finally, the side reaction causes the potential of the electrode to be a mixed (corrosion) potential. It is commonly assumed that the lithium electrode is covered by the SEI layer. However, there is strong evidence that, in many situations, the protection is not complete and side reactions involving the solvent or anion continuously occur. Such reactions can increase the concentration of lithium ions adjacent to a lithium electrode, introducing error into measurements of the variation in potential with apparent electrolyte concentration, particularly at low electrolyte concentrations. Such concentration-ceU measurements are used to obtain activity coefficients and transference numbers via the galvanostatic polarization method. Simulations of the type described in this section can be used to analyze how much error is introduced by the side reaction [73]. It may be preferable to use a less reactive reference electrode, such as Li4Ti50,2 [74], to reduce this error. 

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