Metallic ions, fixed charge

Before the quantum theory of solids (see description in Chapter 21), microscopic descriptions of metals were based on the Drude model, named for the German physicist Paul Drude. The solid was viewed as a fixed array of positively charged metal ions, each localized to a site on the solid lattice. These fixed ions were surrounded by a sea of mobile electrons, one contributed by each of the atoms in the solid. The number density of the electrons, is then equal to the number density of atoms in the solid. As the electrons move through the ions in response to an applied electric field, they can be scattered away from their straight-line motions by collisions with the fixed ions this influences the mobility of the electrons. As temperature increases, the electrons move more rapidly and the number of their collisions with the ions increases therefore, the mobility of the electrons decreases as temperature increases. Equation 22.7 applied to the electrons in the Drude model gives... 

Diffusion is not straightforward inside the resin phase, and this is due to the restrictive influences of the polymer network and because of the charge distribution connected with the fixed ions of the functional groups. The resin phase is consequently related to a porous solid. The effectual diffusivities of metal ions in the resin phase may differ but are largely less than those in the aqueous phase external to the resin phase. If Fick s law is applied to diffusion in a resin bead of radius, r, it may be represented as... 

The picture of the compact double layer is further complicated by the fact that the assumption that the electrons in the metal are present in a constant concentration which discontinuously decreases to zero at the interface in the direction towards the solution is too gross a simplification. Indeed, Kornyshev, Schmickler, and Vorotyntsev have pointed out that it is necessary to assume that the electron distribution in the metal and its surroundings can be represented by what is called a jellium the positive metal ions represent a fixed layer of positive charges, while the electron plasma spills over the interface into the compact layer, giving rise to a surface dipole. This surface dipole, together with the dipoles of the solvent molecules, produces the total capacity value of the compact double layer. 

In fact, this attraction between negative charges (that violates the principles of electrostatics) is mediated by the crystal structure of the superconductor. In every metal lattice there is a reciprocal stripping of valence electrons between metal sites which results in these metal sites, fixed at lattice positions, assuming a positive charge. As shown in Figure 7, when a moving electron crosses these positive metal sites the metal ions are attracted towards the electron trajectory and disturbed from its equilibrium position. 

In the case of cytochrome c, these electrostatic terms are due to changes in the redox states of the internally bound protein metal ion. In other cases where the charges on anions or cations are numerically fixed, the ions can dissociate (e.g., as the metal ion leaves the protein) or migrate (e.g., Na% K+, Ca2+, Cl-, HPO2, H+). If the exchange of these ions involves sites, especially hydrophobic sites, deep inside proteins, on the one hand, and free solution or surface sites, on the other hand, then they will be expected to have an electrostatic influence on the protein much as in a change of redox state. Thus we look next at two calcium binding proteins and later at insulin. 

Equation (4.4.1b) expresses impermeability of the ideally cation-permselective interface under consideration for anions j is the unknown cationic flux (electric current density). Furthermore, (4.4.1d) asserts continuity of the electrochemical potential of cations at the interface, whereas (4.4. lg) states electro-neutrality of the interior of the interface, impenetrable for anions. Here N is a known positive constant, e.g., concentration of the fixed charges in an ion-exchanger (membrane), concentration of metal in an electrode, etc. E in (4.4.1h) is the equilibrium potential jump from the solution to the interior of the interface, given by the expression ... 

Figure 15-13 Ion-exchange equilibria on surfaces of a glass membrane H4 replaces metal ions bound to the negatively charged oxygen atoms. The pH of the internal solution is fixed. As the pH of the external solution (the sample) changes, the electric potential difference across the glass membrane changes.

FIGURE 38 Selective diffusion across ion-exchange membranes. (a) Anion exchange, and (b) cation exchange. Metal cations are designated by M+, anion A-, proton H+, and the fixed charges in the membrane by + and -. 

In ceramics containing transition metal ions the possibility of hopping arises, where the electron transfer is visualized as occurring between ions of the same element in different oxidation states. The concentration of charge carriers remains fixed, determined by the doping level and the relative concentrations in the different oxidation states, and it is the temperature-activated mobility, which is very much lower than in band conduction, that determines a. 

In metallic conductors the metal ions are positively charged and exactly (in solid metals) or more or less precisely (as in liquid metals) fixed by the lattice, but the electrons are free to move, behaving like an "electron gas". Because of their high concentration and high mobility, the conductivity K is very high (10 -10 S m ) and so eu"e the polarizability and the dielectric permittivity, c... 

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