Electroneutrality condition solution

Diffusion in Binary Electrolytes at Nonzero Currents Consider a reaction in which one of the ions of the binary solution is involved. For the sake of definition, we shall assume that its cation is reduced to metal at the cathode. The cation concentration at the surface will decrease when current flows. Because of the electroneutrality condition, the concentration of anions should also decrease under these conditions (i.e., the total electrolyte concentration c. should decrease). 

When the extraction of the hydrophilic counteranion from the aqueous solution into the membrane bulk is negligible (cation permselectivity preserved), the concentration of the complex cation in the membrane bulk Cb, is equal to that of the fixed anionic sites, X, in the membrane matrix, due to the electroneutrality condition within the membrane bulk ... 

The sample solution contains a fixed concentration of supporting electrolyte E" L and a varying concentration of primary salt M X . The ionophore I is confined in the membrane. Only the primary cation can be complexed with the ionophore I (given stoichiometry 1 1 stability constant The complex MI and the anionic site are the lipophilic species that are present only in the membrane phase. In this system, the electroneutrality condition at the membrane bulk leads to... 

The electroneutrality condition decreases the number of independent variables in the system by one these variables correspond to components whose concentration can be varied independently. In general, however, a number of further conditions must be maintained (e.g. stoichiometry and the dissociation equilibrium condition). In addition, because of the electroneutrality condition, the contributions of the anion and cation to a number of solution properties of the electrolyte cannot be separated (e.g. electrical conductivity, diffusion coefficient and decrease in vapour pressure) without assumptions about individual particles. Consequently, mean values have been defined for a number of cases. 

Because of the electroneutrality condition, the individual ion activities and activity coefficients cannot be measured without additional extrather-modynamic assumptions (Section 1.3). Thus, mean quantities are defined for dissolved electrolytes, for all concentration scales. E.g., for a solution of a single strong binary electrolyte as... 

A great many electrolytes have only limited solubility, which can be very low. If a solid electrolyte is added to a pure solvent in an amount greater than corresponds to its solubility, a heterogeneous system is formed in which equilibrium is established between the electrolyte ions in solution and in the solid phase. At constant temperature, this equilibrium can be described by the thermodynamic condition for equality of the chemical potentials of ions in the liquid and solid phases (under these conditions, cations and anions enter and leave the solid phase simultaneously, fulfilling the electroneutrality condition). In the liquid phase, the chemical potential of the ion is a function of its activity, while it is constant in the solid phase. If the formula unit of the electrolyte considered consists of v+ cations and v anions, then... 

It is characteristic for the actual diffusion in electrolyte solutions that the individual species are not transported independently. The diffusion of the faster ions forms an electric field that accelerates the diffusion of the slower ions, so that the electroneutrality condition is practically maintained in solution. Diffusion in a two-component solution is relatively simple (i.e. diffusion of a binary salt—see Section 2.5.4). In contrast, diffusion in a three-component electrolyte solution is quite complicated and requires the use of equations such as (2.1.2), taking into account that the flux of one electrically charged component affects the others. 

The electroneutrality condition can be expressed by the condition of charge balance among the species in solution, according to... 

The region extending from the phase boundary out to about 3 nm is quite unlike the solution beyond. Generalizations valid elsewhere in the solution do not necessarily apply here. In this inner zone, the so-called double-layer region [9], we may encounter a violation of the electroneutrality condition (see Sect. 4.1) and large electric fields. Concentrations may be enhanced or depleted compared with the adjacent solution. 

Defect thermodynamics provide the guidelines for the solution of this practical problem. In Chapter 2, the basic ideas on how to influence point defect concentrations by doping with (heterovalent) additions were presented. Due to the electroneutrality condition and the laws of mass action, we can control the point defect... 

The steps for constructing and interpreting an isothermal, isobaric thermodynamic model for a natural water system are quite simple in principle. The components to be incorporated are identified, and the phases to be included are specified. The components and phases selected "model the real system and must be consistent with pertinent thermodynamic restraints—e.g., the Gibbs phase rule and identification of the maximum number of unknown activities with the number of independent relationships which describe the system (equilibrium constant for each reaction, stoichiometric conditions, electroneutrality condition in the solution phase). With the phase-composition requirements identified, and with adequate thermodynamic data (free energies, equilibrium con-... 

Figure 3 shows the predicted behavior of the pH of the solution as a function of leachant renewal frequency for the same system parameters. As can be seen, the higher the flow rate, the sooner steady state is achieved, and the closer the leachant composition to that of the original solution. In particular, the pH curve for the static case ( = 0) shows that the solution pH has not reached steady state yet after 28-days leaching. Approach to steady state under the static leaching conditions can be a very lengthy process. However, an equilibrium pH value can be estimated by use of the solution electroneutrality condition as applied to the reactions modeled. Indeed, at all times ... 

For simplicity, the model equations are reduced for the description of monovalent salt solutions. In particular, the model is verified for a Na+Cl solution. Thus, with the valences z+ = 1, z = —1 and zJc = —1, an external concentration Cm and an internal concentration cm are defined via the electroneutrality condition of the external and the internal solutions, respectively ... 

Moreover, with the electroneutrality condition (18)2, the equations (16) and (17) can be used to compute the equilibrium concentration and the osmotic pressure of the internal solution [3] ... 

Equation 4.3 is formally similar to a complexation reaction between SR(s) and the aqueous solution species on the left side. Indeed, the solid-phase product on the right side can be interpreted on the molecular level as either an outer-sphere or an inner-sphere surface complex. The latter type of adsorbed species was invoked in connection with the generic adsorption-desorption reactions in Eqs. 3.46 and 3.61, which were applied to interpret mineral dissolution processes. In general, adsorbed species can be either diffuse-layer ions or surface complexes,7 and both species are likely to be included in macroscopic composition measurements based on Eq. 4.2. Equation 4.3, being an overall reaction, does not imply any particular adsorbed species product, aside from its stoichiometry and the electroneutrality condition in Eq. 4.4. 

Distribution (Nernst) potential — Multi-ion partition equilibria at the -> interface between two immiscible electrolyte solutions give rise to a -> Galvanipotential difference, Af(j> = (j>w- 0°, where 0wand cj>°are the -> inner potentials of phases w and o. This potential difference is called the distribution potential [i]. The theory was developed for the system of N ionic species i (i = 1,2..N) in each phase on the basis of the -> Nernst equation, the -> electroneutrality condition, and the mass-conservation law [ii]. At equilibrium, the equality of the - electrochemical potentials of the ions in the adjacent phases yields the Nernst equation for the ion-transfer potential,... 

According to the electroneutrality condition accumulation of ion A becomes practicable owing to diffusion of a large number of coins Y from the solution into the bead (Fig. 3, C, curve II) under the effect of the electric held. 

In many calculations the hydrogen ion concentration is more accessible than the activity. For example, the electroneutrality condition is written in terms of concentrations rather than activities. Also, from stoichiometric considerations, the concentrations of solution components are often directly available. Therefore, the hydrogen ion concentration is most readily calculated from equilibrium constants written in terms of concentration. When a comparison of hydrogen ion concentrations with measured pH values is required (in calculation of equilibrium constants, for example), an estimate of the hydrogen ion activity coeflScient can be made by application of the Debye-Huckel theory if necessary, an estimate of liquid-junction potentials also can be made. Alternatively, the glass electrode can be calibrated with solutions of known hydrogen ion concentration and constant ionic strength. " ... 

Figure 7.10. Models Ila and b. Solubility of MeCOjCs) as a function of pH at constant Pcor Here —log pccb = 3.5 (corresponding to partial pressure of CO2 in atmosphere). If no excess acid or base is added (Cb — Ca = 0), the equilibrium composition of the solution is given by the electroneutrality condition 2[Me ] = [HCOfJ. This condition is indicated by a vertical dash slightly displaced from the intersection of [Me ] with HCOf ]. The inset gives — log[Me ] for pure MeC03(s) suspensions in equilibrium with pco = 10 atm as a function of p/T. ...

Solution of type (b) is characterized by the electroneutrality condition (Figure 15.4b) ... 

Figure 14. (A) Diagram of the charge distribution in the triple layer model. (B) Flat capacitors connected in series as equivalent of a triple layer model at the aqueous solution/metal oxide interface. Charge distribution on capacitor plates is obtained from the electroneutrality condition written in the form 6g = (— o) + ( d).

The electrode equilibria in cell (9.5.35) are exactly the same as those in cell (9.5.23). However, the process at the liquid junction is very different from that at the central Hg Hg2S04 electrode in cell (9.5.23). In order to maintain electroneutrality, both the cation and anion must be involved in the equilibrium at the liquid junction. The process at this junction is complicated by the fact that the individual ions move with different mobilities. An easy way of analyzing the operation of this cell is to consider the cell reactions under reversible conditions. At the left-hand electrode Zn ions enter solution 1 according to reaction (9.5.31). In order to maintain electroneutrality in solution 1, Zn ions move out to solution 2, and SO ions move in the opposite direction from solution 2 to solution 1. If the fraction of the current across the liquid junction carried by the Zn ions is then transport of Zn " " is described as... 

It is worth noting here that the exact solution of a set of nonlinear equations for more complicated equilibria is often unachievable. In such cases, the approximation method implying a simplification of the overall electroneutrality condition using the only pair of predominant defects can be useful. This approach can be illustrated on the basis of the above example of a Si crystal. As the equilibrium constants (Equations (3.15-3.17)) are functions of temperature, the concentrations of different defects can alter in different ways, depending on the value of the pre-exponential factor K° and the enthalpy of the defects reaction, AH . As a result, it is possible to choose a temperature range where the overall electroneutrality condition (Equation (3.18)) can be approximated by pairing the predominant defects. In this case, two possible approximations can be suggested ... 

Point (microscopic) defects in contrast from the macroscopic are compatible with the atomic distances between the neighboring atoms. The initial cause of appearance of the point defects in the first place is the local energy fluctuations, owing to the temperature fluctuations. Point defects can be divided into Frenkel defects and Schottky defects, and these often occur in ionic crystals. The former are due to misplacement of ions and vacancies. Charges are balanced in the whole crystal despite the presence of interstitial or extra ions and vacancies. If an atom leaves its site in the lattice (thereby creating a vacancy) and then moves to the surface of the crystal, it becomes a Schottky defect. On the other hand, an atom that vacates its position in the lattice and transfers to an interstitial position in the crystal is known as a Frenkel defect. The formation of a Frenkel defect therefore produces two defects within the lattice—a vacancy and the interstitial defect—while the formation of a Schottky defect leaves only one defect within the lattice, that is, a vacancy. Aside from the formation of Schottky and Frenkel defects, there is a third mechanism by which an intrinsic point defect may be formed, that is, the movement of a surface atom into an interstitial site. Considering the electroneutrality condition for the stoichiometric solid solution, the ratio of mole parts of the anion and cation vacancies is simply defined by the valence of atoms (ions). Therefore, for solid solution M X, the ratio of the anion vacancies is equal to mJn. 

The conditions (8.49), (8.51), and (8.53), are often referred to as the electroneutrality conditions since in most cases they apply to electrolyte solutions where the total charge in the system is fixed. However, it is clear that the same conditions apply to any solute D that dissociates into two fragments... 

Polarizabilities are simulated also in salt solutions [48], This time the contribution of coions to the electroneutrality condition must be taken into account. We consider that polyion phosphate charge is compensated by the same amount of net charge. At every simulation step we numerically sort all the small ions in increasing order of the sum of their distances from both ends of the polyion. Then if we find a coion in the sorting list, we search a... 

The TPM is generally designated for the chemical stimulation and can provide local chemical as well as mechanical unknowns. In most cases, the gel phase only is investigated by prescribing the concentrations at the boundary of the gel (at the gel-solution interface) by using the Donnan Equation (7) together with the electroneutrality condition of (6). 

When excess supporting electrolyte is present, Laplace s equation can still be used to solve for the tertiary current distribution (mass transport of ions plays a role). In such case the potential field is coupled to the concentration field at the electrode surface [264]. In the more general case of concentration gradients, the solution electroneutrality condition is imposed instead of Laplace s equation (assuming spatially invariant dielectric constant)... 

By the same argument as before, the integral may be taken outside the summations for a symmetrical reference system (e.g. the RPM electrolyte) and applying the electroneutrality condition one sees that in this case Kj = K. Since the first two terms in the SL expansion form an upper bound for the free energy, the limiting law as T oo at constant c must always be approached from one side, unlike the Debye-Huckel limiting law which can be approached from above or below as c 0 at fixed temperature T (e.g. ZnSO and HCl in aqueous solutions). 

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