Total ionic equation defined

The terms in equation 1.166 represent total ionic polarizability, composed of electronic polarizability a plus an additional factor a , defined as a displacement term, due to the fact that the charges are not influenced by an oscillating electric field (as in the case of experimental optical measurements) but are in a static field (Lasaga, 1980) ... 

The above theory can also be applied to account for the concentration dependence of transport numbers, especially in dilute solutions. Since the transport number can be defined as a ratio of the equivalent conductance of the given ion to the total ionic conductance (equation (6.7.6)), it is clear that a non-linear relationship can be derived describing the concentration dependence using equations (6.9.23) and (6.9.24). 

The beginning of the modern approach to strong-electrolyte solutions was G. N. Lewis (1913) empirical observation that y depends approximately only on the total ionic strength, T, of the solution and not on the specific ions present. The total ionic strength is defined by the equation... 

The concentration units of A and N in the aqueous and micellar pseudophases require special consideration. At low micelle concentrations, for example, <3-5% of the total solution volume, a common experimental condition, the molarities of A and N in the aqueous pseudophase can be expressed in stoichiometric units because the micelles occupy only a small fraction of the total pseudophase volume, (12), square brackets. However, the rate of the reaction in the micellar pseudophase depends on then-concentrations in the interfacial region and not on the whole solution volume. This point is crucial. The concentrations of A and N in the interfacial region are expressed in moles/Uter of interfacial volume. Both A and N are assumed to be located in Ihe interfacial region because they are polar or ionic. Equation (13) defines the relationship between the interfacial concentration of N, Nm, and the stoichiometric concentrations of N and S in micelles ... 

If, in the same way, we use (72) to define for the other processes the characteristic units J, L, and Y, similar remarks can be made with regard to J and J, with regard to L and L, and likewise with regard to Y and Y. By equation (72) a precise definition has been given to the characteristic unit of any process and we must hope that in the future the study of ionic solutions will eventually provide a complete interpretation of these quantities. At the present time we are very far from this goal. At any rate the total unitary quantity for each process must be isolated and evaluated before it can be interpreted. In the remaining chapters of this book we shall have occasion to mention only the quantities D, L, Y, J, and U, defined in accordance with (72) and (73). If, however, anyone should wish to give a precise definition to a quantity that includes less than the whole of the unitary term, the symbols in bold-faced type remain available for this purpose. 

Since the ideal solution is defined by the absence of interactions between ions, the total energy WIS required to charge the central ion and its ionic cloud in an ideal solution is obtained from Equation 14 by setting Wint = 0 ... 

The isoelectric point may be conveniently defined as the ZPC expected for a pure, single component solid oxide, hydrous oxide, or hydroxide with a nondefective structure in an electrolyte totally devoid of specifically adsorbed polar or ionic species. An IEP(s) can be calculated from the charge and size of the cation using Equation 13 and the constants in Tables I and II. The maximum accuracy to be expected may be judged from the graphical correlation given in Figure 3. 

The input of the problem requires total analytically measured concentrations of the selected components. Total concentrations of elements (components) from chemical analysis such as ICP and atomic absorption are preferable to methods that only measure some fraction of the total such as selective colorimetric or electrochemical methods. The user defines how the activity coefficients are to be computed (Davis equation or the extended Debye-Huckel), the temperature of the system and whether pH, Eh and ionic strength are to be imposed or calculated. Once the total concentrations of the selected components are defined, all possible soluble complexes are automatically selected from the database. At this stage the thermodynamic equilibrium constants supplied with the model may be edited or certain species excluded from the calculation (e.g. species that have slow reaction kinetics). In addition, it is possible for the user to supply constants for specific reactions not included in the database, but care must be taken to make sure the formation equation for the newly defined species is written in such a way as to be compatible with the chemical components used by the rest of the program, e.g. if the species A1H2PC>4+ were to be added using the following reaction ... 

Equations (3) or (4), with refinements as necessary for "local field" effects, are an appropriate and useful basis for discussion of various models of non-conducting solutions of biological species considered in I. In many cases, however, solutions of interest have appreciable ionic concentrations in the natural solvent medium and the polymer or other solute species may also have net charges. Under these conditions, the electrical response is better considered in terms of the total current density Jfc(t) defined and expressed by linear response theory as... 

In the case of an - electrolyte dissociating in solution as Aj,+ Bj, < is+Az+ + z/ Bz where v+z+ = v z to ensure electroneutrality, and the total number of particles formed by each molecule is v = v+ + z/, then the only activity that can be measured is that of the complete species, and the individual ions cannot be assigned meaningful chemical potentials. Under these circumstances, a mean activity coefficient is defined through the equation yv = y++ yvs. Since individual ionic chemical potentials are not measurable, it has become conventional to assign to the chemical potential of the hydrogen ion under standard conditions the value of zero, allowing relative chemical potentials for all other ions to be formulated. 

Another way to define ionic charges consists in partitioning space into elementary volumes associated to each atom. One method has been proposed by Bader [240,241]. Bader noted that, although the concept of atoms seems to lose significance when one considers the total electron density in a molecule or in a condensed phase, chemical intuition still relies on the notion that a molecule or a solid is a collection of atoms linked by a network of bonds. Consequently, Bader proposes to define an atom in molecule as a closed system, which can be described by a Schrodinger equation, and whose volume is defined in such a way that no electron flux passes through its surface. The mathematical condition which defines the partitioning of space into atomic bassins is thus ... 

Hiickel theory [or the Giintelberg or Davies equation (Table 3.3)] may be used to convert the solubility equilibrium constant given at infinite dilution or at a specified / to an operational constant, valid for the ionic strength of interest. In seawater solubility equilibrium constants, experimentally determined in seawater, may be used. For example, the CaC03 calcite solubility in seawater of specified salinity may be defined by = [Ca " ] [CO f ], where [Caj ] and [C03f ] are the total concentrations of calcium and carbonate ions, for example,... 

Equations (2)-(4) show that the total potential energy of interaction between two colloidal spherical particles depends on the surface potential of the particles, the effective Hamaker constant, and the ionic strength of the suspending medium. It is known that the addition of an indifferent electrolyte can cause a colloid to undergo aggregation. Furthermore, for a particular salt, a fairly sharply defined concentration, called critical aggregation concentration (CAC), is needed to induce aggregation. 

High-temperature ionic solvents are known to contain relatively high total concentrations of cations (e.g. in the KCl-LiCl eutectic, the concentration of Li+ is approximately equal to 8.5 mol kg-1 of the melt). Usually, cation-anion complexes in molten salts are characterized by co-ordination numbers of the order of 4-6. This means that the maximal consumption of acidic cations does not exceed 0.4-0.6 mol kg-1 in diluted solutions with concentrations close to 0.1 mol kg-1. This estimate is considerably lesser than the initial concentration of acidic cations in the pure melt. In the case of the KCl-LiCl eutectic melt, this consumption is only of the order of 5-7%, and the value of NMe+ in equation (1.3.16) may be assumed to be constant. Therefore, for each ionic solvent of the second kind (kind II) the denominator in equation (1.3.16) is a constant which characterizes its acidic properties. We shall define p/L = -log /L to be the relative measure of acidic properties of a solvent and call it the oxobasicity index of ionic melt [37, 162, 181]. Since the direct determination of the absolute concentration of free oxide ions in molten salts is practically impossible, the reference melt should be chosen— for this melt, /L is assumed to be 1 and p/L = 0. The equimolar KCl-NaCl... 

Since statistical mechanical calculations can, in principle, be extended to take account of the interactions between one ion and very many others there is no need to think in terms of the ionic atmospheres of the ions present. There also is no need to invoke the Poisson equation, since what is being calculated is a quantity which is defined by the total potential energy which... 

In the above equations (19.60) to (19.64), G is the total Gibbs free energy of the system jdi is the chemical potential of species i ni is the number of moles of species i N is the total number of species in the system c is the total number of components (for present purposes considered as the elements) Be is the number of moles of each component (or element, e) in the system hei is the number of moles of component (or element) e contained in one mole of species t p refers to a separate electrolyte (ionic) solution phase, and (j> is the total number of these electrolyte phases Zp i refers to the valence or charge of the ith species in the pth phase and is the slack variable defined in (19.64). 

The E and E° values as used in the Nernst equation are defined in terms of the activities of individual species. In very dilute solutions in which there is no formation of complexes, concentration can be used in place of activity and there is no need to account for the portion of the total concentration of a species that might be complexed. In many solutions, however, ionic interactions, complex formation, and acid-base reactions must be taken into account. Formal potentials, E and E are often used for this purpose and apply only to a given set of solution conditions. For example, for the half-reaction... 

An ab initio formulation of VB theory is possible but it is more cumbersome than its MO or DFT equivalents. Instead, the strategy adopted by Warshel has been to define the principal resonance structures for the reaction process of interest and, for each, parametrize an appropriate empirical function that describes the energy of the structure as a function of the relevant geometrical parameters. The ground state potential energy surface is then obtained by solving the secular equation for the resonance structures. Let us take H-F again as our example [21]. In this case, there are two resonance structures, the covalent and the ionic, with wavefimctions V i and respectively. The total wavefunction for the system, is a linear combination of these wavefimctions ... 

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