Equilibria Including Activity Coefficients

In order to demonstrate the power of the Solver in Excel, let us return to the problem mentioned in the introduction to this chapter (p.31) What is the solubility of calcium sulphate but this time taking into account activity coefficients. As it turns out, they are far from zero, even in a saturated solution of only slightly soluble gypsum. 

We need two equations, (a) The ionic strength y of a solution is defined as half of the sum of all products of the concentrations Cj multiplied by the square of their charges zy 

And (b) the extended Debye-Huckel equation for the approximation of the activity coefficient yj of the j-th ion. It needs the charge Zi and the ionic radius ay 

Any alternative approximation for the activity coefficients could be applied, the principle is the same. 

The cell B4 contains a guess for the solubility. This allows the computation of the concentration for both ions in the cells D7 and D8. These in turn define the ionic strength in BIO, computed by applying equation (3.56). Next, 

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Including activity coefficients y, the thermodynamic equilibrium constant K becomes ... 

Equation 8-5 is the rear equilibrium constant. Equation 6-2. the concentration quotient, K,., did not include activity coefficients ... 

Table A.2 is model output for seawater freezing at 253.15 K. Beneath the title, the output includes temperature, ionic strength, density of the solution (p), osmotic coefficient amount of unfrozen water, amount of ice, and pressure on the system. Beneath this line are the solution and gaseous species in the system. The seven columns include species identification, initial concentration, final (equilibrium) concentration, activity coefficient, activity, moles in the solution phase, and mass balance. The mass balance column only contains those components for which a mass balance is maintained. The number of these components minus 1 is generally the number of independent components in the system (in this case, 8 — 1 = 7). The mass balances (col. 7) should equal the initial concentrations (col. 2). This mass balance comparison is a good check on the computational accuracy.

It is important to be aware of certain assumptions that are implicit in the derivation of the above equations. Most importantly, it is assumed that 5C and 8H are themselves independent of concentration and temperature effects. While this can often be experimentally verified for 8H, it is generally impossible to verify for 5C unless the slow-exchange limit can be attained. Another potential error is the failure to include activity coefficients in the equilibrium expressions, even though concentrations often exceed 1 M. Since iterative nonlinear curve fitting often involves locating a relatively shallow minimum, effects such as these can lead to significant error in derived K. 

It is seen, therefore, that if the equilibrium constant of a reaction could be determine experimentally, the standard e.h.f. of the cell in which that reaction takes place can be calculated. The constant K is, of course, the true (thermodynamic) equilibrium constant, and in its determination allowance should be made for departure of the solution from ideal behavior, either by including activity coefficients or by extrapolation to infinite dilution. 

The discussion thus far might lead one to say that the experimental evidence indicates that the initial collision process occurs between the ions. Actually, because of the rapidly established equilibrium [Eq. (7-34)], the ionic mechanism is no more probable on a kinetic basis than the collision process involving neutral molecules. To see why this is so we need only write down the rate equation for both mechanisms, including activity coefficients. For the ionic reactants. 

In the broadest sense, of course, no model is unique (see, for example, Oreskes et al., 1994). A geochemical modeler could conceptualize the problem differently, choose a different compilation of thermodynamic data, include more or fewer species and minerals in the calculation, or employ a different method of estimating activity coefficients. The modeler might allow a mineral to form at equilibrium with the fluid or require it to precipitate according to any of a number of published kinetic rate laws and rate constants, and so on. Since a model is a simplified version of reality that is useful as a tool (Chapter 2), it follows that there is no correct model, only a model that is most useful for a given purpose. 

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... 

The derivative equations for osmotic and activity coefficients, which are presented below, were applied to the experimental data for wide variety of pure aqueous electrolytes at 25°C by Pitzer and Mayorga (23) and to mixtures by Pitzer and Kim (11). Later work (24-28) considered special groups of solutes and cases where an association equilibrium was present (H PO and SO ). While there was no attempt in these papers to include all solutes for which experimental data exist, nearly 300 pure electrolytes and 70 mixed systems were considered and the resulting parameters reported. This represents the most extensive survey of aqueous electrolyte thermodynamics, although it was not as thorough in some respects as the earlier evaluation of Robinson and Stokes (3). In some cases where data from several sources are of comparable accuracy, a new critical evaluation was made, but in other cases the tables of Robinson and Stokes were accepted. 

Very few generalized computer-based techniques for calculating chemical equilibria in electrolyte systems have been reported. Crerar (47) describes a method for calculating multicomponent equilibria based on equilibrium constants and activity coefficients estimated from the Debye Huckel equation. It is not clear, however, if this technique has beep applied in general to the solubility of minerals and solids. A second generalized approach has been developed by OIL Systems, Inc. (48). It also operates on specified equilibrium constants and incorporates activity coefficient corrections for ions, non-electrolytes and water. This technique has been applied to a variety of electrolyte equilibrium problems including vapor-liquid equilibria and solubility of solids. 

Figure 7 shows the predicted vapor-phase mole fractions of HC1 at 25°C as a function of the liquid-phase molality of HC1 for a constant NaCl molality of 3. Also included are predicted vapor-phase mole fractions of HC1 when the interaction parameter A23 is taken as zero. There are unfortunately no experimental vapor-liquid equilibrium data available for the HC1-NaCl-FLO system however, considering the excellent description of the liquid-phase activity coefficients and the low total pressures, it is expected that predicted mole fractions would be within 2-3% of the experimental values. 

Since the acid HX acts as a solvent, its activity may be regarded as constant and included in the equilibrium constant. is the mean activity coefficient of the cation acid and the stabilizing anion X. The way in which equations (5) and (6) are written define the corresponding equilibrium constants as basicity constants K. Their reciprocal corresponds to the acidity constant and gives the acid strength of the conjugate acid AH. ... 

Calculate the temperatures and vapor compositions from the vapor-liquid equilibrium data, using the subroutine BUfiPT. Raoult s law is used in the example, but nonideality can be included by adding activity coefficient equations. Newton-Raphson convergence is used. 

Two chapters on activity coefficients and the systematic treatment of equilibrium from the sixth edition were condensed into Chapter 8. A new, advanced treatment of equilibrium appears in Chapter 13. This chapter, which requires spreadsheets, is going to be skipped in introductory courses but should be of value for advanced undergraduate or graduate work. New topics in the rest of this book include the acidity of metal ions in Chapter 6, a revised discussion of ion sizes and an example of experimental design in Chapter 8. pH of zero charge for colloids... 

If the reaction mixture is very dilute in the reactants and the products, the activity coefficients can all be approximated by unity. Then the last term on the right hand side of Eq. (2.20) vanishes, and the left hand side can be written as AG° = -RT n ATsolution, the equilibrium quotient becoming the equilibrium constant. Under ordinary conditions, however, the activity coefficient term must be taken into account, since there are solvent effects on all the terms on the right hand side except -RTIn K". The fact that different numbers of solvent molecules may specifically associate with the reactants and the products and that solvent molecules may be released or consumed in the reaction should not be included explicitly, since this effect is already covered by the terms in AG s of solvation of the reactants and products according to our definition of this concept. 

In the previous discussion, activity coefficients have been totally neglected, for simplicity, but they should be included in a proper treatment. However, for ionic amphiphiles, where activity corrections are expected to be most important, additional complications arise. If the counterions are explicitly incorporated in the equilibria the number of possible chemical species is greatly enhanced making a detailed analysis even more complex. Furthermore a description of a process in terms of an equilibrium constant is only really suitable when the forces involved are of a short range... 

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 ... 

Select the criterion to be used for thermodynamic consistency. Deviations from thermodynamic consistency arise as a result of experimental errors. Impurities in the samples used for vapor-liquid equilibrium measurements are often the source of error. A complete set of vapor-liquid equilibrium data includes temperature T. pressure P. liquid composition x, and vapor composition y,. Usual practice is to convert these data into activity coefficients by the following equation, which is a rearranged form of the equation that rigorously defines K values (i.e., defines the ratio y, /x, under Related Calculations in Example 3.1) ... 

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