Transference number

The measurement of the conductivity yields the sum of the positive and negative ion conductivities. To obtain the individual ion conductivities, an additional independent measurement is necessary. Even bef ore Kohlrausch demonstrated the law of independent migration of ions, it was commonly supposed that each ion contributed to the flow of current. In 1853 Hittorf devised a method to measure the contribution of the individual ions. 

The transference number of an ion is defined as the fraction of the current carried by that ion. By Eq. (31.29) the conductivity of a solution containing any number of electrolytes is K = CiXi, then by definition the transference number of the fcth ion is 

The transference number of an ion is not a simple property of the ion itself it depends on which other ions are present and on their relative concentrations. It is apparent that the sum of the transference numbers of all the ions in the solution must equal unity. 

In a solution containing only one electrolyte, it follows from Eq. (31.38) that the transference numbers, and, are defined by 

But electrical neutrality in the compound requires that v+ = v z thus we see that 

It has been already said that the conduction of electricity in an electrolyte is due to both anions and cations. The part of the total quantity of electricity transferred by each of the two species respectively depends on the relative velocity of the anions and the cations. Evidence to this can be gained from the following calculations. 

If the equation (III-19) is substituted in the equation (III-10) then the total current flowing through the electrolyte can be determined by using the following equation  

From the total current part /+ is transferred by the cation and part / by the anion (see also III-21)  

Designating the part of the total current transferred by cation as t+ and the part transferred by anion as then dividing the equations (IV-2) an (TV-3) by (IV-l) we get the following important formulae  

The values t+ and ( are called transference numbers of cation and anion the tables contain only the values of + as the value of = 1—1+. By dividing the last two equations we find that the ratio of the transference numbers equals the ratio of mobilities of both ions or the ratio of their ionic conductances. Only if the velocity of both ions is identical will equal quantities of electricity be transferred by both anions and cations. 

Properties such as formation of ion pairs, viscosity, conductivity, and mobility are important factors to describe the efficiency of ion transport. Dielectric relaxation spectroscopy (DRS) [529] is a method that has not been applied for studying electrolytes related to lithium ion batteries. This group anticipates that this situation 

For a completely dissociated electrolyte, the transference number of an ion is the number of faradays of electricity carried by the ion concerned across a reference plane, fixed with respect to the solvent, when one faraday of electricity passes across the plane [436]. 

This simple definition is invalid if the electrolyte is no longer ideal and contains, for example, ion associates. The transference number has to be defined much more generally the transference number of a cation- or anion-constituent R is the net number of faradays carried by that constituent in the direction of the cathode or anode, respectively, across a reference plane fixed with respect to the solvent, when 1 F of electricity passes across the plane [436]. With this definition the transference number tR can also be negative, for example, if an ion A+ is carried in a net transport as an ion-constituent AX2 to the anode. 

To determine transference numbers of liquid electrolytes, generally classical methods are used. These classical methods, meaning moving boundary [436], Hittorf s method, and combined data from emf [437], involve extensive experiments. 

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Because both positive and negative ions move under the influence of an electric field, albeit in opposite directions, the fraction of the current carried by an ion is given by the ratio of the mobiUty of the ion and the total mobiUty of all the ions in solution. This ratio is called the transference number t of the ion. Thus ... 

The fraction of the total conductivity at a specific temperature and composition owing to the conduction of specie i is called the transference number... 

Table 1. Transference Number of Cations, Anions t, and Electrons or Holes in Several Compounds...

Equation 11 gives the conductivity for a particular ion having a transference number, in a crystal, which is the Nemst-Einstein relationship. 

Current flow through the frits is supported by ions. Cations and anions both support the virtual current by flowing in opposite directions, and the transference number of a particular ion is defined as the fraction of the total current it carries. The sum of all transference numbers then is necessarily unity. If the fraction of the virtual current carried by the cations equals the fraction carried by the anions then the solution is said to be equitransferent. 

Both the anions and cations can contribute to the current. In the absence of concentration gradients, the transference number relates the fraction of current carried by each species... 

Cp a = specific heat of air at constant pressure AT jj = temperature rise for stoichiometric combustion D = surface average particle diameter Pa = air density Pf = fuel density

equivalence ratio B = mass transfer number... 

The mass transfer number B represents the ratio of the energy available for vaporization to the energy required for vaporization, and may be thought of as a driving force for mass transfer. It can be expressed as... 

Liquid Hydrocarbon Fuei Mass Transfer Number B... 

Because the ionic transference number for zirconia material is taken as being unity, then this equation reduces to the Nernst equation " ... 

Transporf Number the proportion Of the current carried by a parlicuiar iqn (transfer number)., , , ... 

Despite the results from various experiments such as transference number measurements, polarographic studies, spectroscopic measurements, and dielectric relaxation studies in addition to conductivity measurements, unilateral triple-ions remain a matter of debate. For experimental examples and other hypotheses for the interpretation of conductance minima the reader is referred to Ref. [15] and the literature cited there. 

It is worth mentioning that single-ion conductivities of lithium ions and anions at infinite dilution, and transference numbers of ligand-solvated lithium ions estimated therefrom, increase due to the replacement of more than one (generally four) solvent molecules. Table 6 demonstrates this beneficial feature. 

A 0.2 mol L"1 LiCl/THF solution possesses only very low conductivity of 1.6xl0"6 Scm"1. Addition of N(CH2 CH2NR2)3(R = CF3S02 short nomenclature M6R) yields an increase in conductivity by three orders of magnitude to 1.7 x 10 3 S cm"1. This approach is seemingly especially useful for battery electrolytes, because the transference number of the lithium ion is increased. Conceptually this approach is similar to the use of lith-... 

For a fully dissociated salt, all techniques should give the same values of transport number, t. Transference number measurements are appropriate for electrolytes containing associated species and any technique within one of the three groups will give a similar response, but values of 7] across the groups may vary. 

Electronic conductivity of thin-film solid electrolytes. Besides having low electronic transference numbers, it is essential for thin films of the order of 1 jim that the magnitude of the electronic resistance is low in order to prevent self-discharge of the battery. For this reason, specific electronic resistances in the range of 1012-1014 Qcm are required for thin-film solid electrolytes. Often the color may be a valuable indication of the electronic conductivity. In this regard, solid electrolytes should preferably be transparent white [20]. 

As the cell is discharged, Zn2+ ions are produced at the anode while Cu2+ ions are used up at the cathode. To maintain electrical neutrality, SO4- ions must migrate through the porous membrane,dd which serves to keep the two solutions from mixing. The result of this migration is a potential difference across the membrane. This junction potential works in opposition to the cell voltage E and affects the value obtained. Junction potentials are usually small, and in some cases, corrections can be made to E if the transference numbers of the ions are known as a function of concentration.ee It is difficult to accurately make these corrections, and, if possible, cells with transference should be avoided when using cell measurements to obtain thermodynamic data. 

Migration—movement of charged particles along an electrical field (i.e., the charge is carried through the solution by ions according to their transference number). 

Holz, M Lucas, O Muller, C, NMR in the Presence of an Electric Current, Simultaneous Measurements of Ionic Mobilities, Transference Numbers, and Self-Diffusion Coefficients Using an NMR Pulsed-Gradient Experiment, Journal of Magnetic Resonance 58, 294, 1984. Hooper, HH Baker, JP Blanch, HW Prausnitz, JM, Swelling Equilibria for Positively Ionized Polyacrylamide Hydrogels, Macromolecules 23, 1096, 1990. 

From measurements of conductivities, transfer numbers (electro-migration of charged species), lattice constants and experimental densities, it has been shown that Frenkel defects predominate (Lidlard -1957). This means that ... 

Thus, we measure formation rate in air, pure oxygen gas and then in an inert gas. If the rates do not differ significantly, then we can rule out gaseous transport mechanisms. There are other tests we can apply, including electriccd conductivity, transference numbers and thermal expcmsion. Although these subjects have been investigated in detail, we shall not present them here. 

Selvaratnam, M. Spiro, M. (1965). Transference numbers of orthophosphoric acid and the limiting equivalent conductance of the HgPO ion in water at 25 °C. Transactions of the Faraday Society, 61, 360-73. 

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