Solution activity and activity coefficients

THE MATHEMATICS OF FUGACITY, IDEAL SOLUTIONS, ACTIVITY AND ACTIVITY COEFFICIENTS... 

The derivation of the various relations involving fugacity, ideal solutions, activity, and activity coefficients covers several pages with mathematics. Professors and graduate students enjoy that, but most undergraduates do not. For that reason most of that mathematics is placed in this appendix, so that the discussion in Chapter 7 can flow more easily. The pertinent results of that mathematics are transferred from here to Chapter 7. 

For gases, pure solids, pure liquids, and nonionic solutes, activity coefficients are approximately unity under most reasonable experimental conditions. For reactions involving only these species, differences between activity and concentration are negligible. Activity coefficients for ionic solutes, however, depend on the ionic composition of the solution. It is possible, using the extended Debye-Htickel theory, to calculate activity coefficients using equation 6.50... 

Many additional consistency tests can be derived from phase equiUbrium constraints. From thermodynamics, the activity coefficient is known to be the fundamental basis of many properties and parameters of engineering interest. Therefore, data for such quantities as Henry s constant, octanol—water partition coefficient, aqueous solubiUty, and solubiUty of water in chemicals are related to solution activity coefficients and other properties through fundamental equiUbrium relationships (10,23,24). Accurate, consistent data should be expected to satisfy these and other thermodynamic requirements. Furthermore, equiUbrium models may permit a missing property value to be calculated from those values that are known (2). 

In an attempt to explain the nature of polar interactions, Martire et al. [15] developed a theory assuming that such interactions could be explained by the formation of a complex between the solute and the stationary phase with its own equilibrium constant. Martire and Riedl adopted a procedure used by Danger et al. [16], and divided the solute activity coefficient into two components. 

The net retention volume and the specific retention volume, defined in Table 1.1, are important parameters for determining physicochemical constants from gas chromatographic data [9,10,32]. The free energy, enthalpy, and. entropy of nixing or solution, and the infinite dilution solute activity coefficients can be determined from retention measurements. Measurements are usually made at infinite dilution (Henry s law region) in which the value of the activity coefficient (also the gas-liquid partition coefficient) can be assumed to have a constant value. At infinite dilution the solute molecules are not sufficiently close to exert any mutual attractions, and the environment of each may be considered to consist entirely of solvent molecules. The activity... 

Many reactions encountered in extractive metallurgy involve dilute solutions of one or a number of impurities in the metal, and sometimes the slag phase. Dilute solutions of less than a few atomic per cent content of the impurity usually conform to Henry s law, according to which the activity coefficient of the solute can be taken as constant. However in the complex solutions which usually occur in these reactions, the interactions of the solutes with one another and with the solvent metal change the values of the solute activity coefficients. There are some approximate procedures to make the interaction coefficients in multicomponent liquids calculable using data drawn from binary data. The simplest form of this procedure is the use of the equation deduced by Darken (1950), as a solution of the ternary Gibbs-Duhem equation for a regular ternary solution, A-B-S, where A-B is the binary solvent... 

Although nearly identical solid-aqueous solution compositions are observed in recrystallization from two directions under conditions of total constant composition, this alone is insufficient proof of the establishment of equilibrium. In order to test for equilibrium, the solid solution activity coefficients must be determined and used to compare observed solid and aqueous solution compositions with the appropriate values expected at equilibrium. 

Using the experimental solution compositions (Table IV) and the calculated aqueous solution activity coefficient ratio Ygr-/Y( - (Table VI), Figure 4 shows the slopes of log K... 

Figure 17.11. Extrapolation of relative activity coefficients to obtain 1 /y2 for the calculation of solute activity coefficients. Data from Ref. 4. Dimethyl sulfoxide is the solvent, and water is the solute.

Membrane deterioration may be merely caused by decrease of acetyl content(C ) in the active surface layer as a result of hydrolysis or oxidation, not by structure change. Analysis was carried out based on solution-diffusion model proposed by Lonsdale etal( ), using their measured values of solute diffusivity and partition coefficient in homogeneous membrnaes of various degree of acetyl content and also using those values of asymmetric membranes heat treated at various temperatures measured by Glueckauf(x) ... 

I. 46. The magnitude of the coefficient reflects the electric charge distribution of the ionic species. A 0.1 molal solution of Al2(S04)3 has an activity coefficient of only 0.035. It should also be noted that, in dilute solutions, activity coefficients of electrolytes decrease in magnitude with increasing concentration. A minimum is reached and the coefficient then increases with concentration. See Activity Debye-Huckel Law Biomineralization... 

We define the standard state of a liquid as ay = 1 and for gases as an ideal gas pressure of 1 bar, Pj = I- For ideal liquid solutions (activity coefficients of unity), we write ay = Cy so at chemical equilibrium... 

In many studies of the SLE of ILs, three methods have been used to derive the solute activity coefficients, pj, from the so-called correlation equations that describe the Gibbs free energy of mixing (GE), fhe Wilson [103], UNIQUAC ASM [104], and NRTLl [105] models. Historically, fhe UNIQUAC... 

For most of the systems with alcohols, the description of SLE was given by the average standard mean deviation (oj) < 2 K for UNIQUAC ASM and NRTL 1 equations. The procedure of correlation has been described in many articles [52-54,79,84-88,91-94]. Using GE models the solute activity coefficients in the saturated solution, y, were described. 

For infinitely dilute solutions activity coefficients approach unity so the activity and the concentration of an ion will be equal. For calculations involving more concentrated solutions corrections must be made using activity coefficients, especially when relating the calculated concentration of species to an imposed mass (mole) balance constraint. The activity coefficients can be calculated from a number of ion activity theories and the relevant equations for some of the commonly used ones are shown below. 

In addition to knowing the TP dependence of equilibrium constants (Eqs. 2.25 and 2.28), we must also know the T-P dependence of solute activity coefficients and the osmotic coefficient of the solution. A theoretical model, such as Pitzer s approach, is necessary for this purpose because activity coefficients and the osmotic coefficient must be defined at finite concentrations and not simply for the infinitely dilute state, which suffices for equilibrium constants (Eqs. 2.25 and 2.28). 

The chemical equilibrium model, FREZCHEM, requires calculation of solute activity coefficients (7) and the osmotic coefficient ((f)) in concentrated solutions (Chap. 3). In this work, the Pitzer approach is used to calculate these quantities. 

A fundamental concept in all theories for determining activity coefficients is that ionic interactions are involved. These interactions cause a deviation in the free energy associated with the ions from what it would be if they did not occur. Consequently, at the limit of an infinitely dilute solution, activity coefficients go to 1 because there are no ionic interactions. This basic consideration also leads to the idea that as the concentration of ions increases, their extent of interaction must also increase. Ionic strength is a measure of the overall concentration of ions in a solution and the fact that more highly charged ions exert a greater influence on ionic interactions. It is calculated as ... 

The standard state for solutes in the (HL) reference is therefore the hypothetical state of pure solute (x, = 1), but with solute molecules interacting only with solvent molecules (y, = 1). Practically, chemical potentials in the standard state are obtained by making measurements at very low concentrations and extrapolating them to X,- = 1, assuming that Henry s law continues to hold to this concentration. At nonzero concentration of solutes, activity coefficients in the (HL) reference measure deviations of the solution from ideally dilute behavior. 

The first term on the right side of this equation is the capacity factor, at zero ionic strength. The ratio of activity coefficients in the second term is the ratio of the solute activity coefficients at finite ionic strength over the solute activity coefficient at zero ionic strength. The activity coefficients of the mobile phase modifier are modeled by the Deye-Hiickel theory and can be approximated by... 

Concentration and activity of a solute are only the same for very dilute solutions, i.e. yi approaches unity as the concentration of all solutes approaches zero. For non-dilute solutions, activity coefficients must be used in chemical expressions involving solute concentrations. Although freshwaters are sufficiently dilute to be potable (containing less than about 1000 mg total dissolved solids (TDS)), it cannot be assumed that activity coefficients are close to unity. 

For solutions of fixed ionic strength, or, for example, where major ions in solution, e.g. conservative cations and anions, are present at concentrations several orders of magnitude greater than the species involved in the chemical equilibrium, e.g. A, B, C, and D in Equation (3.13), it can be assumed that the solute activity coefficients are also constants and can be incorporated into the equilibrium constant. The equilibrium constant for a fixed ionic strength aqueous solution is termed a constant ionic strength equilibrium constant, K. 

From a practical viewpoint we may conclude that molecular solutes have activity coefScients near unity up to an ionic strength of 0.1 and that deviations are moderate even at ionic strengths of the order of unity. In contrast to those of ionic solutes, activity coefficients of molecular solutes usually are slightly greater than unity. 

Equation (7-4) indicates that the solubility product includes an activity-coefficient term, a term which has been assumed to be unity up to this time. The introduction to this chapter pointed out that errors arising from neglect of the effects of the activity coefficient are usually small when compared with several uncertainties or side reactions. The activity coefficient in Equation (7-4) depends on the kind and concentration of all electrolytes in solution, not merely those involved directly with the precipitate. The correction to solubility calculations that must be made to account for the activity-coefficient effect is known as the diverse ion effect. The appropriate background is discussed in Chapter 2, and Problems 2-1,2-2, and 2-3 are examples of the calculations. For 1 1 electrolytes in solution, activity coefficients can usually be assumed to be unity when concentrations are much less than 0.1 M. Common ion and diverse ion effects can be significant at the same time, for example, when a large excess of common ion is added in a precipitation. The diverse ion effect is one of the reasons that the haphazard addition of a large excess of precipitant should be avoided. 

Many nonionizable organic solutes in water are described thermodynamically on the mole fraction scale, although their solubilities may commonly be reported in practical units, for example, molality. [Refer to Schwarzenbach et al. (1993) and Klotz (1964) for detailed discussion of such aqueous solutions.] Here, the standard state is the pure liquid state of the organic solute, that is, Xj = 1. The reference state is Xi - 1, that is, a solution in which the organic solute molecules interact with one another entirely. Activity coefficients of solute molecules in dilute aqueous solutions are generally much greater than unity for this reference state choice, jc, 1. For example, with this reference state, aqueous benzene has an experimental infinitely dilute solution activity coefficient, T nzeno of 2400 for an infinite dilution reference state, jc, - 0, the activity coefficient would be approximately 1 (Tanford, 1991). 

X4, and y2,q are the mole fractions of components 2—4 and the solute activity coefficient, respectively, in the quaternary mixture. The above derivative under isothermal—isobaric conditions could be obtained from those of the chemical potential with respect to the number of particles ... 

Activity coefficient models are equations representing the Gibbs or the Helmholtz energy of solutions. Activity coefficients and related properties are derived form these energy functions by proper differentiation (Equation (1)). 

An alternative method of data reduction was reported early in the history of gas chromatography by Hoare and Purnell (12-15 see refs. 16,17 for recent applications), who considered the dependence oTffie specific retention volume on the solute saturation vapor pressure pA. Thus, taking the view [now recognized to be naive (18) see later], that the observed mole fraction-based solute activity coefficient" can be decomposed into "athermal" and "thermal" components (19-22) v v-... 

As presented above, and in most of the published literature, the isotherm model equations are expressed in terms of total sorbate concentrations. In MINTEQA2, however, aqueous species activities are used rather than their concentrations. This is preferable in that reactions are written in terms of activities. Also, so that the concentrations of sorption sites, sorbed species, and dissolved species are computationally equivalent, the former are entered in MINTEQA2 as moles of sites per liter of solution. Activity coefficients of sorption sites and sorbed species are generally taken to be equal to unity by most authors and in MINTEQA2. 

In our discussion of aqueous species of iron, it is appropriate to first consider the Fe(II) and Fe(III) hydroxyl complexes (Table 12.1). To judge their importance, we will construct plots to show the fractional distribution of these complexes as a function of pH. The computational approach is as presented in Chap. 3. Solute activity coefficients are ignored. 

---