Electroneutrality approximation

Fig. 2.2. Relation between the Na+ and CI concentrations of geothermal waters and of inclusion fluids. The solid line indicates the condition of electroneutrality approximated by the equation = mc -. Solid and open circles mean the chemical analytical data on inclusion fluids and geothermal waters, respectively. S = Salton Sea R = Reykjanes W = Wairakei B = Broadlands O = Otake H = Hveragerdi C = Climax D = Darwin P = Providencia (Shikazono, 1978a).

Therefore, a detailed analysis of the conformations of many-armed star-like PEs may take place within the so-called local electroneutrality approximation (LEA). The latter also allows for an equilibrium of the ionization of the macroion, hence also covering the pH-responsive (annealing) star-like PEs. 

The local electroneutrality approximation (LEA) assumes that the local number density of charged monomer units in the PE corona is approximately equal to local (excess) number density of mobile counterions ... 

The mean-field approach provides a convenient framework for the analysis of copolymer self-assembly leading to micellar structures. Combining the mean-field approach with the local electroneutrality approximation (LEA) enables us to generalize the theory for micelles with ionic coronal blocks that feature stimuli-responsive properties. 

Structural rearrangements in a micelle with quenched PE corona can be investigated using the mean-field approach (described in Sect. 5) in combination with the local electroneutrality approximation (LEA). As before, we neglect here the radial gradients in the polymer density and mobile ion distributions (i.e., implement the boxlike model). Moreover, we omit the contribution due to nonelectrostatic... 

The mean-field approach combined with the local electroneutrality approximation (LEA) can be extended beyond the boxlike model. To calculate the free energy of a micellar corona with the radial gradients in polymer density, we assume that all the free ends of blocks A are located at the corona periphery and are equally stretched. Within this model, the free energy of a micelle of morphology i (t = 1,2,3), can be explicitly calculated for both quenched and annealing PE coronae. We outline here only a general scheme of the approach, whereas the details can be found in [22, 32]. 

Oldham KB, Bond AM (2001) How valid is the electroneutrality approximation in the theory of steady-state voltammetry J Electroanal Chem 508 28 0... 

Historically, another common choice for a further constraint has been the electroneutrality approximation. This uses the small absolute value of cq to approximate that ... 

This is called the DHLL+52 approximation. On carrying out the integrations over q y(r) and q iryll and using the electroneutrality condition, this can be rewritten as [63]... 

The compound Lajln has Tc = 10.4 K. Because La is hypoelectronic and In is hyperelectronic, I expect electron transfer to take place to the extent allowed by the approximate electroneutrality principle.13 The unit cube would then consist of 2 La, La, and In+, with In+ having no need for a metallic orbital and thus having valence 6 with the bonds showing mainly pivoting resonance among the twelve positions. The increase in valence of In and also of La (to 3 f ) and the assumption of the densely packed A15 structure account for the decrease in volume by 14.3%. Because the holes are fixed on the In + atoms, only the electrons move with the phonon, explaining the increase in Tc. 

The electrochemical potential of single ionic species cannot be determined. In systems with charged components, all energy effects and all thermodynamic properties are associated not with ions of a single type but with combinations of different ions. Hence, the electrochemical potential of an individual ionic species is an experimentally undefined parameter, in contrast to the chemical potential of uncharged species. From the experimental data, only the combined values for electroneutral ensembles of ions can be found. Equally inaccessible to measurements is the electrochemical potential, of free electrons in metals, whereas the chemical potential, p, of the electrons coincides with the Fermi energy and can be calculated very approximately. 

As noted already, the first approximation of electroneutrality relation is... 

As illustrated in Fig. 5, large errors on logP are produced when approximated by Eq. (14). Indeed, in the pH region where the ionized species dominate, the proportion of neutral molecules in the aqueous phase becomes rapidly negligible. The ions are then liable to partition into the organic phase provided they form an ion pair with an electrolyte present in the aqueous phase or a species of equal charge crosses the interface in the opposite direction in order to maintain electroneutrality. 

Because it is assumed that Ks is a lot greater than Ke it is reasonable to ignore the minority electronic defects and approximate the electroneutrality Eq. (7.12) by the relation ... 

This central, medium pressure, region of the diagram displays the electroneutrality equation approximation [V J = [Vj J at the top. In this region the compound... 

The low-pressure region displays the electroneutrality equation approximation [e ] = 2[Vx ]. Electrons predominate so that the material is an n-type semiconductor in this regime. In addition, the conductivity will increase as the partial pressure of the gaseous X2 component decreases. The number of nonmetal vacancies will increase as the partial pressure of the gaseous X2 component decreases, and the phase will display a metal-rich nonstoichiometry opposite to that in the high-pressure domain. Because there is a high concentration of anion vacancies, easy diffusion of anions is to be expected. 

At ever higher partial pressures the number of nonmetal vacancies will diminish more and more and the number of holes will exceed the number of vacancies in this regime. Thus the approximate electroneutrahty Eq. (7.13) will no longer be representative. A more appropriate form of the electroneutrality equation for the high-pressure region is obtained by ignoring the minority species, nonmetal vacancies and electrons, to give... 

The approximations to use depend upon the pressure regime and the values of the equilibrium constants. This oxide is an insulator under normal conditions, and so, in the middle region of the diagram, Schottky equilibrium is dominant, that is, > Ke and the electroneutrality equation is approximated by... 

In the oxidized region, holes and cation vacancies are preferred, so the electroneutrality equation is approximated by... 

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