Activity coefficients Davies equation

Fig. 8.1. Activity coefficients y, predicted at 25 °C for a singly charged ion with size a of 4 A, according to the Debye-Huckel (Eqn. 8.2), Davies (Eqn. 8.4), and B-dot (Eqn. 8.5) equations. Dotted line shows the Davies equation evaluated with a coefficient of 0.2 instead...

Activity coefficients in the aqueous phase, yiw, of neutral molecules are set equal to one because of the zero charge, and under the assumption that the activity coefficient of the infinitely diluted solution equals the actual activity coefficient. The activity coefficients of the charged species can be approximated with the Davies equation ... 

TLM Activity Coefficients. In the version of the TLM as discussed by Davis et al. (11), mass action equations representing surface complexation reactions were written to include "chemical" and "coulombic" contributions to the overall free energy of reaction, e.g., the equilibrium constant for the deprotonation reaction represented by Equation 2 has been given as... 

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. 

Various empirical relations are available for calculating individual ion activity coefficients [discussed by Stumm and Morgan (1996) for natural waters and Sposito (1984a, b), for soil solutions]. In the calculations in this book I used the Davies equation ... 

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

Fig. 39. Activity coefficients calculated from the data of Fig. 37 and from an additional set of data obtained from a duplicate run compared with the Gibbs-Duhem equation. Reprinted with permission from Allen, T. M., Taflin, D. C, and Davis, E. J., Ind. Eng. Chem. Res. 29, 682-690, Copyright 1990, American Chemical Society.

The second semiempirical approach to the evaluation of the activity coefficient is the use of the Davies equation which modifies the Bronsted extension of the limiting Debye-Hilckel expression and is given by... 

Figure 13-2 Activity coefficients from extended Debye-Huckel and Davies equations. Shaded areas give Debye-Huckel activity coefficients for the range of ion sizes in Table 8-1.

Even if we know all reactions and equilibrium constants for a given system, we cannot compute concentrations accurately without activity coefficients. Chapter 8 gave the extended Debye-Huckel equation 8-6 for activity coefficients with size parameters in Table 8-1. Many ions of interest are not in Table 8-1 and we do not know their size parameter. Therefore we introduce the Davies equation, which has no size parameter ... 

For ionic species, we compute activity coefficients with the Davies equation 13-18. For the neutral species H,P04, we assume that y 1.00. 

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. 

Mixture of KH2P04 and Na2HP04 including activity coefficients from Davies equation ... 

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 ... 

Input values of concentration, volume, and moles are in cells B3 B6 in Figure 13-13. Cell B7 has the value 2 to indicate that glycine is a diprotic acid. Cell B8 has the activity coefficient of H+ computed with the Davies equation, 13-18. Cell B9 begins with the effec-... 

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. 

B. IB Repeat Exercise 13-A with activity coefficients from the Davies equation. 

D. 11 Include activity coefficients from the Davies equation to find the pH and concentrations of species in the mixture of sodium tartrate, pyridinium chloride, and KOH in Section 13-1. Consider only Reactions 13-1 through 13-4. 

From pAj and pA 2 for glycine at p, = 0 in Table 10-1, compute p/fj and pA 2, which apply at p, = 0.1 M. Use the Davies equation for activity coefficients. Compare your answer with experimental values in cells B10 and Bll of Figure 13-13. 

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.

Finally, there is the matter of ion activity coefficients. The Davies equation given in Chapter 1 will be used, because all of the solutions are in the dilute range. In addition, ion pairing corrections will be made for the CaHCC>3+ and CaC03° ion pairs. This step requires iteration in calculation of the ion activity coefficients. The sequence demanded by the problems is that the concentrations must be initially calculated using ion activity coefficients from the previous case. These new concentrations are then used to calculate new ion activity coefficients, and the process is repeated until the desired degree of precision is reached. Neutral species will be assumed to have an ion activity coefficient of 1, and the ion activity coefficients of H+, OH-, and CaHCC>3+ will be assumed equal. 

Case 1. A raindrop of pure water equilibrates with atmospheric CO2. Let a raindrop of pure water with a pH = 7 (At = 0) form and come to equilibrium with CO2 in the atmosphere at PCO2 = 330 patm. We assume that there is no Ca in the initial water. Also, At cannot change because a neutral species (CO2) is being added. The first thing that must be done is to establish clearly the conditions before the CO2 enters the raindrop. From the pH being equal to 7, it follows that aH+ = 10 7 and aoH = (Kw / aH+) = 10 7. Because H+ and OH- are not involved in ion pairing reactions, from the Davies equation their ion activity coefficients are equal. Therefore, their concentrations will also be equal. 

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. 

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