Electro-neutrality

The transfer of an element from the metal to the slag phase is one in which the species goes from the charge-neutralized metallic phase to an essentially ionic medium in the slag. It follows that there must be some electron redistribution accompanying the transfer in order that electro-neutrality is maintained. A metallic atom which is transfened must be accompanied by an oxygen atom which will absorb the elecuons released in the formation of tire metal ion, thus... 

Consider the crystal, AgBr. Both cation and anion are monovalent, i.e.- Ag+ and Br-. The addition of a divalent cation such as Cd2+ should introduce vacancies, VAg, into the crystal, because of the charge-compensation mechanism. To maintain electro-neutrality, we prefer to define the system as ... 

To maintain electro-neutrality, the following equation is applicaable ... 

For (184)—(186), since these are neutral ligands the uptake of a metal cation is also accompanied by anion binding such that electro-neutrality is maintained. Thus resins of this type may sometimes be useful for separation of either cations or anions. 

In this manner one can see that the terminal monomers of the chain acquire net ionic character (because they can participate in CT only in a single direction), whereas interior monomers remain relatively electro neutral (due to there being equal numbers of CT interactions in and out ). 

A binary ionic solution must contain at least three kinds of species. One example is a solution of AC and BC. Here we have two cation species A+ and B+ and one common anion species C . The sum of the charge of the cations and the anions must be equal to satisfy electro-neutrality. Hence NA+ + NB+ = N(. = N where NA+, AB+ and Nc are the total number of each of the ions and N is the total number of sites in each sub-lattice. The total number of distinguishable arrangements of A+ and B+ cations on the cation sub-lattice is M/N A, JVg+ . The expression for the molar Gibbs energy of mixing of the ideal ionic solution AC-BC is thus analogous to that derived in Section 9.1 and can be expressed as... 

The salt or vinegar acts as an electrolyte, and is needed since the product Al3+ requires counter ions to ensure electro-neutrality (so aluminium ethanoate forms). The oxide ions combine with protons from the vinegar to form water. Figure 7.2 illustrates these processes occurring in schematic form. 

The process described is referred to as ion-exclusion as discussed by Asher and Simpson 9. The resins used are normal and the non-ionic molecules are assumed to be small enough to enter the pores. When large non-ionic molecules are involved, an alternative process called ion-retardation may be used, as discussed by Hatch et al. W]. This requires a special resin of an amphoteric type known as a snake cage poly electrolyte. The polyelectrolyte consists of a cross-linked polymer physically entrapping a tangle of linear polymers. For example, an anion exchange resin which is soaked in acrylic acid becomes entrapped when the acrylic acid is polymerised. The intricacy of the interweaving is such that counter-ions cannot be easily displaced by other counter-ions. On the other hand, ionic mobility within the resin maintains the electro-neutrality. The ionic molecule as a... 

The so-called diffusion layer is still a region dominated by an unequal charge distribution (i.e. in such a zone the principle of electro-neutrality is not valid) due to the electron transfer processes occurring at the electrode surface. In fact, the electrode acts as an electrostatic pump for species of... 

An additional electrostatic component to the polymer interaction term is typically unimportant since the counterions strongly screen any Coulomb interactions [92]. Finally, an electrostatic interaction between polymers and counterions Tint occurs if the PE brush is not locally electro-neutral throughout the system, an example is depicted in Fig. 10a. This energy is given by... 

The exchange reactions take place on the basis of equivalency in accordance with the principle of electro neutrality. The number of milimoles of an ion sorbed by an exchange should correspond to the number of milimoles of an equally charged ion that has been released from the ion exchange [16]. 

Among the basic concepts to be introduced are ionic equilibrium, local equilibrium, local electro-neutrality, etc. 

According to (1.9c) the smallness of suggests that the entire space is divided into the bulk where the local electro-neutrality relations... 

The above estimates suggest that, with the scaling (1.7) valid, it takes about time td for an ionic system to restore local electro-neutrality on the macroscopic length scale and to reach local equilibrium. By local equilibrium we merely imply a state with the normalized ionic fluxes ji, defined by... 

Summarizing, at equilibrium the entire ED cell is divided into the locally electro-neutral bulk solution at zero potential and the locally electroneutral bulk cat- (an-) ion-exchange membrane at ipm < 0 (> 0) potential. These bulk regions are connected via the interface (double) layers, whose width scales with the Debye length in the linear limit and contracts with the increase of nonlinearity. 

We will discuss next the ambipolar diffusion, that is, electro-diffusion of two oppositely charged ions in a solution of a univalent electrolyte with local electro-neutrality. Assume the dimensionless ionic diffusivities are constant. Then the relevant version of (1.9) is... 

Equation (1.57a) implies that in the locally electro-neutral ambipolar diffusion concentration of both ions evolves according to a single linear diffusion equation with an effective diffusivity given by (1.57b). Physically, the role of the electric field, determined from the elliptic current continuity equation... 

A somewhat similar situation occurs in one-dimensional multi-ionic systems with local electro-neutrality in the absence of electric current. It will be shown in Chapter 3 that in this case again the electric field can be excluded from consideration and the equations of electro-diffusion are reduced to a coupled set of nonlinear diffusion equations. 

Finally, we make a terminological remark. In a one-dimensional locally electro-neutral system the expression (1.11b) for the electric current density reduces to... 

The next level is that of one-dimensional electro-diffusion with local electro-neutrality in the absence of an electric current. This is the realm of nonlinear diffusion to be treated in Chapter 3. A still higher level of the same hierarchy is formed by the nonlinear effects of stationary electric current, passing in one-dimensional electro-diflFusion systems with local electro-neutrality. A few typical phenomena of this type will be studied in Chapter 4. The treatment of Chapter 4 will lay the foundation for the discussion of the effects of nonequilibrium space charge characteristic of the fourth level to be treated in Chapter 5. 

Locally Electro-Neutral Electro-Diffusion Without Electric Current... 

Preliminaries. In this chapter we shall address the simplest nonequilibrium situation—one-dimensional locally electro-neutral electrodiffusion of ions in the absence of an electric current. We shall deal with macroscopic objects, such as solution layers, ion-exchangers, ion-exchange membranes with a minimum linear size of the order of tens of microns. 

As pointed out in the Introduction, it is customary in the treatment of such systems to assume local electro-neutrality (LEN), that is, to omit the singularly perturbed higher-order term in the Poisson equation (1.9c). Such an omission is not always admissible. We shall address the appropriate situations at length in Chapter 5 and partly in Chapter 4. We defer therefore a detailed discussion of the contents of the local electro-neutrality assumption to these chapters and content ourselves here with stating only that this assumption is well suited for a treatment of the phenomena to be considered in this chapter. 

This concludes the proof of the above assertion regarding stability of the locally electro-neutral equilibrium and the way it is approached by the system. 

Here C is the concentration vector and D(C) is the diffusivity tensor defined by (3.1.15a). Thus, locally electro-neutral electro-diffusion without electric current is exactly equivalent to nonlinear multicomponent diffusion with a diffusivity tensor s being a rational function of concentrations of the charged species. 

For simplicity diffusivities of both counterions have been assumed equal.) By electro-neutrality... 

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