Selectivity coefficient definition

Selectivity coefficients are not generally constant over the whole exchange isotherm since their definition incorporates concentrations rather than activities. The relation between the thermodynamic exchange constant (/Cth) the mass action constant on a particular concentration scale is obtained by introducing activity coefficients into the expression for the selectivity coefficient, thus ... 

With an arbitrary definition of KNaX as equal to unity, thus establishing a reference half reaction, the equilibrium constant for any other half reaction can be determined from measured selectivity coefficients. The Gapon equation can be readily implemented in this manner. Implementation of the Vanselow equation, however, requires modification of the general equilibrium models to account for the more complex dependence of mole fractions on the molar concentrations. An example ion-exchange calculation using the half reaction approach to represent the Gapon equation is presented in Appendix 2. 

A mathematical description of the retention of ions under gradient elution conditions was introduced in 1957 by Schwab et al. [131]. It is based on parameters which are derived from the normal chromatographic elution process for which the eluent composition is kept constant during the separation. Hence, the retention of an ion at isocratic elution may be described according to Eq. (86), taking into account the definitions for the capacity factor, k, and the selectivity coefficient, K, [see Eq. (35) and (36) in Section 3.2] ... 

Figure 2. The selectivity coefficient,

Interferences can restrict the measurement of ion concentrations, if the interfering ions are highly concentrated or if their selectivity coefficients are high (see Equation [10.26]). For instance, in the preceding case, is about 100 for a glass membrane and if the pH of the analyzed solution is 7, the Na+ detection limit (see Figure 10.7 for the definition of this parameter) is about 10 M. If the pH is 5, this limit is about 10 M. 

Results of models for the Tournemire shales are reported in Table 4. These models were based upon the mineralogical and CEC data presented in Tables 1 and 2 the calculated cation exchange selectivity coefficients and the concentrations of leachable Cl present in the rocks. Preliminary selectivity coefficients (Ac(Mg/Na) and Kc Ca./ Na) close to 3 and 4 respectively) have been derived from aqueous extraction experiments. They are in rather good agreement with literature data (Baeyens Bradbury 1994), and are mainly related to the illite and inter-stratified illite/smectite contents. The leachable Cl has been extracted with pure water at a liquid/solid ratio of 10mlg. It is clear that the Tournemire porewater cannot be definitively described with the data available. The porewater is of Na-Cl-(HCO ) type, with a low salinity, equivalent to a total dissolved... 

The described method can generate a first-order backward or a first-order forward difference scheme depending whether 0 = 0 or 0 = 1 is used. For 9 = 0.5, the method yields a second order accurate central difference scheme, however, other considerations such as the stability of numerical calculations should be taken into account. Stability analysis for this class of time stepping methods can only be carried out for simple cases where the coefficient matrix in Equation (2.106) is symmetric and positive-definite (i.e. self-adjoint problems Zienkiewicz and Taylor, 1994). Obviously, this will not be the case in most types of engineering flow problems. In practice, therefore, selection of appropriate values of 6 and time increment At is usually based on trial and error. Factors such as the nature of non-linearity of physical parameters and the type of elements used in the spatial discretization usually influence the selection of the values of 0 and At in a problem. 

Thus, the value of a definite integral depends on the limits a, b, and any selected variable coefficients in the func tion but not on the dummy variable of integrations. Symbolically... 

SCREEN allows for the selection of urban or rural dispersion coefficients. The urban dispersion option is selected by entering a U (lower or upper case) in column 1, while the rural dispersion option is selected by entering an R (upper or lower case) in column 1. Determination of the applicability of urban or rural dispersion is based upon land use or population density. In general, if 50 percent or more of an area 3 km around the source satisfies the urban criteria (Auer, 1978), the site is deemed in an urban setting. Of the two methods, the land use procedure is considered more definitive. 

For a substance in a given system the chemical potential gi has a definite value however, the standard potentials and activity coefficients have different values in these three equations. Therefore, the selection of a concentration scale in effect determines the standard state. 

Thermodynamics describes the behaviour of systems in terms of quantities and functions of state, but cannot express these quantities in terms of model concepts and assumptions on the structure of the system, inter-molecular forces, etc. This is also true of the activity coefficients thermodynamics defines these quantities and gives their dependence on the temperature, pressure and composition, but cannot interpret them from the point of view of intermolecular interactions. Every theoretical expression of the activity coefficients as a function of the composition of the solution is necessarily based on extrathermodynamic, mainly statistical concepts. This approach makes it possible to elaborate quantitatively the theory of individual activity coefficients. Their values are of paramount importance, for example, for operational definition of the pH and its potentiometric determination (Section 3.3.2), for potentiometric measurement with ion-selective electrodes (Section 6.3), in general for all the systems where liquid junctions appear (Section 2.5.3), etc. 

It has been emphasized repeatedly that the individual activity coefficients cannot be measured experimentally. However, these values are required for a number of purposes, e.g. for calibration of ion-selective electrodes. Thus, a conventional scale of ionic activities must be defined on the basis of suitably selected standards. In addition, this definition must be consistent with the definition of the conventional activity scale for the oxonium ion, i.e. the definition of the practical pH scale. Similarly, the individual scales for the various ions must be mutually consistent, i.e. they must satisfy the relationship between the experimentally measurable mean activity of the electrolyte and the defined activities of the cation and anion in view of Eq. (1.1.11). Thus, by using galvanic cells without transport, e.g. a sodium-ion-selective glass electrode and a Cl -selective electrode in a NaCl solution, a series of (NaCl) is obtained from which the individual ion activity aNa+ is determined on the basis of the Bates-Guggenheim convention for acr (page 37). Table 6.1 lists three such standard solutions, where pNa = -logflNa+, etc. 

The last item has been taken into account by Fujiwara et al. [1980] who introduced a total selectivity similar to Kaiser s selectivity (Eq. 7.21) but products Sj -c instead of the partial sensitivities. However, this definition range is (—1... + oo). A more acceptable range of selectivity results from Doerffel s coefficient of selectivity CSA (Doerffel et al. [1986]) ... 

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