Wagner factors

The expression d In a/d In c is known as the thermodynamic factor and is a special case of the Wagner factor (or thermodynamic enhancement factor) which plays an important role for the kinetic properties of electrodes. This term indicates the deviation from ideality of the mobile component. For ideal systems this quantity becomes 1 and comparison with Pick s first law yields... 

This expression is the general Wagner factor which includes the influence of all the motion of the other species on the motion of species i by the effect of the internal electric fields. W may be larger than 1 which indicates an enhancement of the motion by the simultaneous motions of other species, or W may be smaller than 1 which means that the species are slowed down because of the immobility of other species which are therefore unable to compensate for the electrical charges. The first situation is desirable for electrodes whereas the second one is required for electrolytes in which mobile species should not move except when electrons are provided through the external circuit. Since the transference numbers in Eqn (8.27) include the partial and total conductivities (tj = OjlYjk or the products of the diffusivities (or mobilities) and the concentrations, Eqn (8.27) shows that W depends both on kinetic... 

Compared to Pick s first law for ideal systems (which real systems only approach at the high dilution), two extra terms are introduced (i) the transference number of the electrons and (ii) the Wagner factor. Their part in the electrode kinetics will be discussed in more detail. 

In order to gain some insight into the ionic and electronic mobilities and concentrations that are required for a favourable electrode performance, the simplifying case of constant y and (Henry s or Raoult s law) is considered. Under these conditions the Wagner factor reads according... 

An example of a material (Li3Sb) with a very large Wagner factor is shown in Fig. 8.3. The effective chemical diffusion coefficient is compared with the diffusivity as a function of non-stoichiometry. These data were determined by electrochemical techniques (see Section 8.5). An increase of the diffusion coefficient is observed at about the ideal stoichiometry which corresponds to a change in the mechanism from a predominantly vacancy to interstitial mechanism. The Wagner factor W is as large as 70 000 at the ideal stoichiometry. This gives an effective diffusion coefficient which is more typical of liquids than solids. It is a common... 

Fig. 8.2 The Wagner factor fF as a function of the ratios of the mobilities u and concentrations c of the electronic and mobile ionic species (logarithmic scale).

Fig. 8.3 The Wagner factor, chemical diffusion coefficient and diffusivity of Li in LijSb as a function of stoichiometry at 400 °C. The Wagner factor is as large as 70000 at ideal stoichiometry.

The slope of the coulometric titration curve is accordingly proportional to the Wagner factor in the case of predominant electronic conductivity. 

As discussed in Section 8.2 the relation between the chemical diffusion coefficient and diffusivity (sometimes also called the component diffusion coefficient) is given by the Wagner factor (which is also known in metallurgy in the special case of predominant electronic conductivity as the thermodynamic factor) W = d n ajd In where A represents the electroactive component. W may be readily derived from the slope of the coulometric titration curve since the activity of A is related to the cell voltage E (Nernst s law) and the concentration is proportional to the stoichiometry of the electrode material ... 

The right terms in the two last equations, namely (dpB/dcB z ) and (31n ttBz /31ncBz ), are both called the thermodynamic or -> Wagner factors [iii, iv]. The first of them can be determined experimentally from the ion concentration dependence on the chemical potential of neutral species (a-T-S diagram). The direct determination of the second factor is impossible as the chemical potentials of charged species cannot be explicitly separated from those of other components of a system this parameter can be assessed indirectly, analyzing activities of all components. 

See also - ambipolar conductivity, -> diffusion determination in solids, - Wagner factor, - insertion electrodes, -> batteries. 

The equilibrium conditions in electrochemical systems are usually expressed in terms of electrochemical potentials. For non-equilibrium systems, the gradient of chemical and/or electrochemical potential is a driving force for flux of particles i. See also Wagner equation, - Wagner factor and ambipolar conductivity, -+ On-sager relations. 

Wagner factor — or thermodynamic factor, denotes usually the - concentration derivative of -> activity or - chemical potential of a component of an electrochemical system. This factor is necessary to describe the - diffusion in nonideal systems, where the - activity coefficients are not equal to unity, via Fick s laws. In such cases, the thermodynamic factor is understood as the proportionality coefficient between the selfdiffusion coefficient D of species B and the real - diffusion coefficient, equal to the ratio of the flux and concentration gradient of these species (chemical diffusion coefficient DB) ... 

Concepts of local equilibrium and charged particle motion under - electrochemical potential gradients, and the description of high-temperature -> corrosion processes, - ambipolar conductivity, and diffusion-controlled reactions (see also -> chemical potential, -> Wagner equation, -> Wagner factor, and - Wagner enhancement factor). 

The chemical diffusion coefficient D is the product of the diffusivity of the ions Dj and the Wagner factor S na.J r lncv, [3],... 

The thermodynamic factor 3 In aiou/3 In (which is a special case of the Wagner factor, as described later) is sometimes very large and enhances the ionic flitx above that which would be expected from the concentration gradient alone. In a predominantly electronic conductor in which the concentrations of electrons or holes are very large, i.e., the chemical potentials of the electronic species are essentially uniform throughout the material, the gradients of the chemical potentials of neutral atoms and their respective ions are identical. The transport of ions may be considered to be the same as the net transport of neutral species in this case. 

Wagner Factor which Relates the Chemical Diffusion Coefficient to the Diffusivity D ( D =WD) for Various Conditions. 

The values of the Wagner factor under various conditions are sununarized in Table 9.1. This term of thermodynamic quantities can be interpreted kinetieally as the generation of internal electric fields by two differently mobile species. If one kind, e.g., electrons, has a significantly higher mobility than the other kind, e.g., ions, the more mobile species will tend to move ahead of the other ones. The reqnirement for overall eharge flux neutrality causes the more mobile species to be slowed down and the less mobile ones to be speeded up. ... 

---