Equilibrium distribution of ions

The only potential that varies significantly is the phase boundary potential at the membrane/sample interface EPB-. This potential arises from an unequal equilibrium distribution of ions between the aqueous sample and organic membrane phases. The phase transfer equilibrium reaction at the interface is very rapid relative to the diffusion of ions across the aqueous sample and organic membrane phases. A separation of charge occurs at the interface where the ions partition between the two phases, which results in a buildup of potential at the sample/mem-brane interface that can be described thermodynamically in terms of the electrochemical potential. At interfacial equilibrium, the electrochemical potentials in the two phases are equal. The phase boundary potential is a result of an equilibrium distribution of ions between phases. The phase boundary potentials can be described by the following equation ... 

This is the equilibrium distribution of ions before excitation, which is evidently the eigenfunction of L. 

We assume that the equilibrium distribution of ions obeys a Boltzmann distribution so that volume charge density p r) at position r, r being the distance from the particle center (r > a), is given by... 

The concentration-based equilibrium distribution of ions in this exchange reaction is defined by... 

Since electrical neutrality must be maintained in ion-exchange chromatography, the distribution of ions between the mobile and stationary phases is complicated by the need for counterions. The use of partition or distribution ratios in the usual way is therefore not entirely appropriate instead, a selectivity coefficient is used to denote the equilibrium distribution of ions. Thus for (25-1) and (25-2) the law of mass action... 

A kinetic argument can be used instead to establish the equilibrium distribution of ions at the semiconductor surface. The rate of adsorption of a species i is given according to Eq. (29) as... 

To demonstrate the general partitioning behavior, the equilibrium distribution of ions is derived in the following discussion under the... 

The selectivity of a particular molecular sieve for a given ion as a function of exchanger composition is normally measured fi om an ion exchange isotherm, which is an isonormal [45], isothermal and reversible plot of equilibrium distributions of ions between the solution and zeohte phases. It is emphasised that it is only vahd to calculate selectivity coeffidents, and derived thermodynamic data, from isotherms which are reversible (that is, the forward and reverse isotherms coindde within experimental uncertainty). The types of isotherms. 

Donnan membrane equilibrium This concerns the distribution of ions on each side of a membrane separating two portions of a solution of... 

It is important to recognize the approximations made here the electric field is supposed to be sulficiently small so that the equilibrium distribution of velocities of the ions is essentially undisturbed. We are also assuming that the we can use the relaxation approximation, and that the relaxation time r is independent of the ionic concentration and velocity. We shall see below that these approximations break down at higher ionic concentrations a primary reason for this is that ion-ion interactions begin to affect both x and F, as we shall see in more detail below. However, in very dilute solutions, the ion scattering will be dominated by solvent molecules, and in this limiting region A2.4.31 will be an adequate description. 

Adsorption and ion exchange share so many common features in regard to apphcation in batch and fixed-bed processes that they can be grouped together as sorption for a unified treatment. These processes involve the transfer and resulting equilibrium distribution of one or more solutes between a fluid phase and particles. The partitioning of a single solute between fluid and sorbed phases or the selectivity of a sorbent towards multiple solutes makes it possible to separate solutes from a bulk fluid phase or from one another. 

Winzor and coworkers have employed measurements of the Donnan distribution of small ions in dialysis equilibrium [14] to reinforce earlier evidence of charge-screening effects in polysaccharide anions [164,165]. These researchers used the absorption optical system of a Beckman XL-1 ultracentrifuge to monitor the distribution of ions in polysaccharide solutions... 

A question of practical interest is the amount of electrolyte adsorbed into nanostructures and how this depends on various surface and solution parameters. The equilibrium concentration of ions inside porous structures will affect the applications, such as ion exchange resins and membranes, containment of nuclear wastes [67], and battery materials [68]. Experimental studies of electrosorption studies on a single planar electrode were reported [69]. Studies on porous structures are difficult, since most structures are ill defined with a wide distribution of pore sizes and surface charges. Only rough estimates of the average number of fixed charges and pore sizes were reported [70-73]. Molecular simulations of nonelectrolyte adsorption into nanopores were widely reported [58]. The confinement effect can lead to abnormalities of lowered critical points and compressed two-phase envelope [74]. 

This thermodynamic equation defines the equilibrium distribution of all permeating ions between the two phases. For quantitative calculations, the conditions of electroneutrality of the phases must be taken into account in addition to this equation. 

From the equilibrium requirement that the chemical potential involving all ionic species be uniform throughout the phase boundary, the distribution of ions within the electrical double layer can be expressed by the Boltzmann equation ... 

Cfs-butene should lead initially to the anti form trrms-butene should lead initially to the syn form and 1-butene should give rise initially to both. The equilibrium distribution of syn and anti forms usually differs greatly from the equilibrium distribution of cis- and frans-butene for cobalt complexes 59, 60) the syn form, precursor of irans-butene, is by far the most stable. By way of contrast for the corresponding carbanion, the cis anion seems by far the more stable. This preference for the cis carbanion is presumed to be the source of the high initial cis-to-trans ratio in the initial products of base catalyzed isomerization. In the base catalyzed isomerization of more complex cf-s-olefins (cfs-S-methyl-stilbene), the ions corresponding to syn and anti are not interconvertible and cis-trans isomeriza-... 

The equilibrium constant for this reaction depends on the stability constants of the ionophore-M+ complexes and on the distribution of ions in aqueous test solution and organic membrane phases. For a membrane of fixed composition exposed to a test solution of a given pH, the optical absorption of the membrane depends on the ratio of the protonated and deprotonated indicator which is controlled by the activity of M+ in the test solution (H,tq, is fixed by buffer). By using a to represent the fraction of total indicator (Ct) in the deprotonated form ([C]), a can be related to the absorbance values at a given wavelength as... 

If the concentration of the diffusible electrolyte is small, relative to that of the colloid electrolyte, only a small quantity of salt will diffuse across the membrane, whilst if the electrolyte is present in relatively large quantities, there will obtain practically an equal distribution on either side of the membrane. A similar treatment permits of the calculation of the equilibrium distribution of a salt containing no ion in common with that of the colloid electrolyte. 

So far, we have used the Maxwell equations of electrostatics to determine the distribution of ions in solution around an isolated, charged, flat surface. This distribution must be the equilibrium one. Hence, when a second snrface, also similarly charged, is brought close, the two surfaces will see each other as soon as their diffuse double-layers overlap. The ion densities aronnd each surface will then be altered from their equilibrinm valne and this will lead to an increase in energy and a repulsive force between the snrfaces. This situation is illustrated schematically in Fignre 6.12 for non-interacting and interacting flat snrfaces. 

The difference between Equations (55) and (60) may be qualitatively understood by comparing the results with the Donnan equilibrium discussed in Chapter 3. The amphipathic ions may be regarded as restrained at the interface by a hypothetical membrane, which is of course permeable to simple ions. Both the Donnan equilibrium (Equation (3.85)) and the electroneutrality condition (Equation (3.87)) may be combined to give the distribution of simple ions between the bulk and surface regions. As we saw in Chapter 3 (e.g., see Table 3.2), the restrained species behaves more and more as if it was uncharged as the concentration of the simple electrolyte is increased. In Chapter 11 we examine the distribution of ions near a charged surface from a statistical rather than a phenomenological point of view. 

For equilibrium conditions, Donnan ion exchange theory (6) may be used to describe quantitatively equilibria for trace quantities of chemical elements in the presence of high homologous salt concentrations. The radionuclides 90Sr and 137Cs constitute a very small fraction of the mass of the chemical constituents in rainfall thus the equilibrium distribution of these nuclides between solid and liquid phases may be estimated as... 

This equation describes an equilibrium distribution of the electroinactive ion in an electrostatic field. This is a typical situation to which one may apply the Boltzman distribution law, which states... 

Summarizing these facts, the equilibrium distribution of the ion pair intermediates can be shifted by the choice of solvent or solvent mixture, which stabilizes either CIP of benzophenone and amine (in acetonitrile) or SSIP (in ethanol). 

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