Ionic strength Davies equation

The Davies (1962) equation is a variant of the Dcbyc-I Iiickcl equation (Eqn. 8.2) that can be carried to somewhat higher ionic strengths. The equation follows from Equation 8.2 by noting that at 25 °C the product a, B is about one. Including an empirical term 0.3 / to the correlation gives,... 

As can be seen in Figure 8.1, the Davies equation does not decrease monotoni-cally with ionic strength, as the Debye-Huckel equation does. Beginning at ionic strengths of about 0.1 molal, it deviates above the Debye-Huckel function and at about 0.5 molal starts to increase in value. The Davies equation is reasonably accurate to an ionic strength of about 0.3 or 0.5 molal. 

Helgeson (1969 see also Helgeson and Kirkham, 1974) presented an activity model based on an equation similar in form to the Davies equation. The model, adapted from earlier work (see Pitzer and Brewer, 1961, p. 326, p. 578, and Appendix 4, and references therein), is parameterized from 0°C to 300 °C for solutions of up to 3 molal ionic strength in which NaCl is the dominant solute. The model takes it name from the B-dot equation,... 

None of these extensions has been really satisfactory and they are not very useful at high ionic strength. The Davies equation (19) differs from the others in providing an additional term which alters the response of the activity coefficient to changes in ionic strength, particularly at higher values. The authors have had some success with this type of equation by replacing the. 2 factor in the second term with a variable. The variable can be determined by experiment at a particular set of conditions. 

The Davies equation [29] has been used extensively to calculate activity coefficients of electrolytes at fairly low ionic strengths. 

The equation has no theoretical foundation but is found to work fairly well up to ionic strengths of 0.1 mol kg It should not be used at higher ionic strengths. The Davies equation has a form similar to the B-G-S equation but with ion interaction coefficients equal to 0.153zf, i.e., 0.15, 0.61, and 1.38 for ions of charge 1, 2, and 3, respectively. These values do not agree very well with the tabulated s values. 

IONIC BOND IONIC RADIUS IONIC STRENGTH ISOTONIC BUFFERS DAVIES EQUATION DEBYE-HOCKEL THEORY Ion-ion interactions,... 

By considering Boi as a single adjustable parameter, data can then be fitted using only Boi and B as variables. A further simplification is to make Boi equal to unity and add a further constant ionic strength term which gives the Davies equation (Davies 1962)... 

Figure 13-3 puts everything together in a spreadsheet. Input values for FKH,P04, FNaiHPOj, pA i, pKn, pK3, and pA w are in the shaded cells. We guess a value for pH in cell H15 and write the initial ionic strength of 0 in cell Cl9. Cells A9 H10 compute activities with the Davies equation. With pi = 0, all activity coefficients are 1. Cells A13 H16 compute concentrations. [HT] in cell B13 is (10 PH)/yH = (10A-H15)/B9. Cell El 8 computes the sum of charges. 

For simplicity, we omit activity coefficients, but you do know how to use them. You would solve the problem with all activity coefficients equal to 1, find the ionic strength, and then compute activity coefficients with the Davies equation. Then you would compute effective equilibrium constants incorporating activity coefficients and solve the problem again. After each iteration, you would find a new ionic strength and a new set of activity coefficients. Repeat the process until ionic strength is constant. Wow You are smart ... 

To use activity coefficients, first solve the equilibrium problem with all activity coefficients equal to unity. From the resulting concentrations, compute the ionic strength and use the Davies equation to find activity coefficients. With activity coefficients, calculate the effective equilibrium constant K for each chemical reaction. K is the equilibrium quotient of concentrations at a particular ionic strength. Solve the problem again with K values and find a new ionic strength. Repeat the cycle until the concentrations reach constant values. 

In this particular example, the Davies equation (15) gives the same value as Equation 1 it seems to be applicable, therefore, to calculating TKs values of oxides and carbonates of bivalent metals from Ks data in 0.2M NaC104. It fails, however, if applied to data which were evaluated in media of higher ionic strengths. 

For modelling purposes the Davis equation is often preferred because only a knowledge of the ionic strength of the medium and the charge on the species are required to calculate the value of yf. As the activities of neutral species, e.g. H4Si05 and Al(OH)3, are not always unity (Pankow, 1991), the activities of these species are calculated using the expression developed by Helgeson (1969) ... 

Ca2+, Al3+, and the Davis equation (dashed line) for ions of different charges (z) as a function of ionic strength. 

Figure 5.2 The relative proportions (a,) of monomeric hydroxy-aluminium species as a function of pH. For the calculation ionic strength was fixed at 0.001 moldmT3 and activity corrections were made using the Davis equation.

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 ... 

These parameters can be directly related back to the information contained in the EPM with n components (j) and m species (t). Application of the mass balance constraint equation requires that the concentration of each species must be known. Therefore, activity coefficients are computed if the ionic strength is already known from either the Davis or the extended Debye-Eluckel equation however, if ionic strength is unknown and has to be calculated, equation (5.134) can be converted to a general expression for the concentration of each species by substituting the expression for S to give... 

In dilute solutions two equations for simple ion activity coefficients as a function of ionic strength have been commonly used. The Davies equation is the simpler of the two and contains no adjustable parameters for different ions. It is ... 

Figure 1.2. The change in the value of the ion activity coefficient as a function of ionic strength for -1 and +2 ions calculated from the Davies equation, and HC03 and Ca2+ calculated using the Debye-Hiickel method.

The BMREP and SDM currently use the Davies technique for activity coefficient prediction. The Davies technique is a combination of the extended Debye-Huckel equation (6) and the Davies equation (7). The Davies technique (and hence both equilibrium models) is accurate up to ionic strengths of 0.2 molal and may be used for practical calculations up to ionic strengths of 1 molal (8). Ion-pair equilibria are incorporated for species that associate (e.g., 1-2 and 2-2 electrolytes). The activity coefficients (y ) are calculated as a simple function of ionic strength (I) and are represented as ... 

The LCM is a semi-theoretical model with a minimum number of adjustable parameters and is based on the Non-Random Two Liquid (NRTL) model for nonelectrolytes (20). The LCM does not have the inherent drawbacks of virial-expansion type equations as the modified Pitzer, and it proved to be more accurate than the Bromley method. Some advantages of the LCM are that the binary parameters are well defined, have weak temperature dependence, and can be regressed from various thermodynamic data sources. Additionally, the LCM does not require ion-pair equilibria to correct for activity coefficient prediction at higher ionic strengths. Thus, the LCM avoids defining, and ultimately solving, ion-pair activity coefficients and equilibrium expressions necessary in the Davies technique. Overall, the LCM appears to be the most suitable activity coefficient technique for aqueous solutions used in FGD hence, a data base and methods to use the LCM were developed. 

Show that for the Davies equation [Equation (25)], y may he greater than 1. Sketch the dependence of y on ionic strength for the Davies equation. 

For charged species it was shown by van der Venne et al. [345] and by Otto and Wegscheider [346] that the so-called Davies equation can be used to describe the effect of the ionic strength on solute retention quantitatively. This equation reads... 

The second term in the DAVIES and extended DEBYE-HUCKEL equations forces the activity coefficient to increase at high ionic strength. This is owed to the fact, that ion interactions are not only based on Coulomb forces any more, ion sizes change with the ionic strength, and ions with the same charge interact. 

Although naturally occurring brines and some high ionic strength contaminated waters may require the more complicated expressions developed in the Davies, SIT, or Pitzer models, the use of Equations (3.3)-(3.5) is justified for the ionic strengths of many freshwaters. 

Values of K, the thermodynamic association constants are given at 25°. The concentrations of ionic species in the solutions at any time can be determined from mass balance, electroneutrality, and the appropriate equilibrium constants as described previously (19, p, 85-92) by successive approximations for the ionic strength. The activity coefficients of Z-valent ionic species may be calculated from an extended form of the Debye-Huckel equation such as that proposed by Davies (20, p. 34-53). 

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