Activity coefficient from solubility measurement

Activity Coefficients from Solubility Measurements.—The activity coefficient of a sparingly soluble salt can be determined in the presence of other electrolytes by making use of the solubility product principle. In addition to the equations already given, this principle may be stated in still another form by introducing the definition of the mean ionic concentration, i.e., c , which is equal to c+clr, into equation (109) this equation then becomes... 

Figure 5.6 Illustration of the effect of a completely water-miscible solvent (CMOS, i.e., methanol) on the activity coefficient of organic compounds in water-organic solvent mixtures decadic logarithm of the activity coefficient as a function of the volume fraction of methanol. Note that the data for naphthalene (Dickhut et al., 1989 Fan and Jafvert, 1997) and for the two PCBs (Li and Andren, 1994) have been derived from solubility measurements whereas for the anilins (Jayasinghe etal., 1992), air-water partition constants determined under dilute conditions have been used to calculate y,f.

It is important to realize, however, that the determination of the substrate-micelle binding constant from solubility data relies entirely on data for saturated solutions and that, in the case of ionic surfactants, differences in the counterion interactions with the micelle and the micelle-substrate complex and activity coefficient effects may seriously complicate the results. In these respects, distribution studies with varying substrate and surfactant concentrations are certainly preferable. In view of the assumptions involved in the derivation and application of equations (10) and (11), the agreement between the K values obtained from kinetic data (equation 10) and those obtained from solubility measurements (equation 11) for several substrate-micelle interactions is certainly both remarkable and significant. 

The activity coefficients of thallous chloride, in the presence of various added electrolytes, determined from solubility measurements at 25 , are recorded in Table XXXII 1 m is the total molality of the thallous chloride and the added substance. It will be observed that in the more dilute solutions the activity coefficient at a given molality is independent of the nature of the other electrolyte present in the solution. The significance of this fact will be considered below. 

To calculate from solubility measurements values of the activity coefficient of thallous iodate in its saturated solution in the presence of potassiiun chloride and of potassium thiocj ate. 

Equations (3-6) for the potentiometric and spectrophotometric methods will provide thermodynamic pKa values. For the solubility-pH dependence method [Eqs. (7-8)], the values obtained are apparent values (pKg )/ which are relevant to the ionic strength (7) of the aqueous buffers used to fix the pH value for each solution. If the ionic strength of each buffer solution is controlled or assessed, then the apparent value can be corrected to a thermod)mamic value, using an activity coefficient from one of the Debye-Hiickel equations (Section 2.2.5). If the solubility-pH dependence is measured in several buffer systems, each with a different ionic strength, then the Guggenheim approach can be used to correct the result to zero ionic strength [Eq. (17)]. 

RandaU. M. C.F. Falley. "The activity coefficient of non-electrolytes in aqueous salt solutions from solubility measurements. The salting-out order of the ions". Chem. Rev., v4, 3. pp285-290 (1927)... 

In principle the activity coefficients yb of solute substances B in a solution can be directly determined from the results of measurements at ven temperature of the pressure and the compositions of the liquid (or solid) solution and of the coexisting gas phase. In practice, this method fails unless the solutes have volatilities comparable with that of the solvent. The method therefore usually fails for electrolyte solutions, for which measurements of ye in practice, much more important than for nonelectrolyte solutions. Three practical methods are available. If the osmotic coefficient of the solvent has been measured over a sufficient range of molalities, the activity coefficients /b can be calculated the method is outlined below under the sub-heading Solvent. The ratio yj/ys of the activity coefficients of a solute B in two solutions, each saturated with respect to solid B in the same solvent but with different molalities of other solutes, is equal to the ratio m lm of the molalities (solubilities expressed as molalities) of B in the saturated solutions. If a justifiable extrapolation to Ssms 0 can be made, then the separate ys s can be found. The method is especially useful when B is a sparingly soluble salt and the solubility is measured in the presence of varying molalities of other more soluble salts. Finally, the activity coefficient of an electrolyte can sometimes be obtained from e.m.f. measurements on galvanic cells. The measurement of activity coefficients and analysis of the results both for solutions of a single electrolyte and for solutions of two or more electrolytes will be dealt with in a subsequent volume. Unfortunately, few activity coefficients have been measured in the usually multi-solute solutions relevant to chemical reactions in solution. 

The Change of Solubility with Temperature. The solubilities of various salts have been measured in aqueous solution at various temperatures. But from these measurements we cannot derive values of L as a function of temperature, until the activity coefficients in the various saturated solutions have been accurately measured. In dilute solutions... 

In the multimedia models used in this series of volumes, an air-water partition coefficient KAW or Henry s law constant (H) is required and is calculated from the ratio of the pure substance vapor pressure and aqueous solubility. This method is widely used for hydrophobic chemicals but is inappropriate for water-miscible chemicals for which no solubility can be measured. Examples are the lower alcohols, acids, amines and ketones. There are reported calculated or pseudo-solubilities that have been derived from QSPR correlations with molecular descriptors for alcohols, aldehydes and amines (by Leahy 1986 Kamlet et al. 1987, 1988 and Nirmalakhandan and Speece 1988a,b). The obvious option is to input the H or KAW directly. If the chemical s activity coefficient y in water is known, then H can be estimated as vwyP[>where vw is the molar volume of water and Pf is the liquid vapor pressure. Since H can be regarded as P[IC[, where Cjs is the solubility, it is apparent that (l/vwy) is a pseudo-solubility. Correlations and measurements of y are available in the physical-chemical literature. For example, if y is 5.0, the pseudo-solubility is 11100 mol/m3 since the molar volume of water vw is 18 x 10-6 m3/mol or 18 cm3/mol. Chemicals with y less than about 20 are usually miscible in water. If the liquid vapor pressure in this case is 1000 Pa, H will be 1000/11100 or 0.090 Pa m3/mol and KAW will be H/RT or 3.6 x 10 5 at 25°C. Alternatively, if H or KAW is known, C[ can be calculated. It is possible to apply existing models to hydrophilic chemicals if this pseudo-solubility is calculated from the activity coefficient or from a known H (i.e., Cjs, P[/H or P[ or KAW RT). This approach is used here. In the fugacity model illustrations all pseudo-solubilities are so designated and should not be regarded as real, experimentally accessible quantities. 

In principle, Gibbs free energies of transfer for trihalides can be obtained from solubilities in water and in nonaqueous or mixed aqueous solutions. However, there are two major obstacles here. The first is the prevalence of hydrates and solvates. This may complicate the calculation of AGtr(LnX3) values, for application of the standard formula connecting AGt, with solubilities requires that the composition of the solid phase be the same in equilibrium with the two solvent media in question. The other major hurdle is that solubilities of the trichlorides, tribromides, and triiodides in water are so high that knowledge of activity coefficients, which indeed are known to be far from unity 4b), is essential (201). These can, indeed, be measured, but such measurements require much time, care, and patience. 

For the purpose of this case study we will select Isopropyl alcohol as the crystallization solvent and assume that the NRTL-SAC solubility curve for Form A has been confirmed as reasonably accurate in the laboratory. If experimental solubility data is measured in IPA then it can be fitted to a more accurate (but non predictive) thermodynamic model such as NRTL or UNIQUAC at this point, taking care with analysis of the solid phase in equilibrium. As the activity coefficient model only relates to species in the liquid phase we can use the same model with each different set of AHm and Tm data to calculate the solubility of the other polymorphs of Cimetidine, as shown in Figure 21. True polymorphs only differ from each other in the solid phase and are otherwise chemically identical. 

The review of Martynova (18) covers solubilities of a variety of salts and oxides up to 10 kbar and 700 C and also available steam-water distribution coefficients. That of Lietzke (19) reviews measurements of standard electrode potentials and ionic activity coefficients using Harned cells up to 175-200 C. The review of Mesmer, Sweeton, Hitch and Baes (20) covers a range of protolytic dissociation reactions up to 300°C at SVP. Apart from the work on Fe304 solubility by Sweeton and Baes (23), the only references to hydrolysis and complexing reactions by transition metals above 100 C were to aluminium hydrolysis (20) and nickel hydrolysis (24) both to 150 C. Nikolaeva (24) was one of several at the conference who discussed the problems arising when hydrolysis and complexing occur simultaneously. There appear to be no experimental studies of solution phase redox equilibria above 100°C. 

Second, the activity coefficient of NaCl has a clear interpretation it provides a quantitative measure of how HC1 changes the solubility of NaCl from its standard-state value of x ... 

Values of the activity coefficients are deduced from experimental data of vapor-liquid equilibria and correlated or extended by any one of several available equations. Values also may be calculated approximately from structural group contributions by methods called UNIFAC and ASOG. For more than two components, the correlating equations favored nowadays are the Wilson, the NRTL, and UNIQUAC, and for some applications a solubility parameter method. The fust and last of these are given in Table 13.2. Calculations from measured equilibrium compositions are made with the rearranged equation... 

ACTIVITY COEFFICIENT. A fractional number which when multiplied by the molar concentration of a substance in solution yields the chemical activity. This term provides an approximation of how much interaction exists between molecules at higher concentrations. Activity coefficients and activities are most commonly obtained from measurements of vapor-pressure lowering, freezing-point depression, boiling-point elevation, solubility, and electromotive force. In certain cases, activity coefficients can be estimated theoretically. As commonly used, activity is a relative quantity having unit value in some chosen standard state. Thus, the standard state of unit activity for water, dty, in aqueous solutions of potassium chloride is pure liquid water at one atmosphere pressure and the given temperature. The standard slate for the activity of a solute like potassium chloride is often so defined as to make the ratio of the activity to the concentration of solute approach unity as Ihe concentration decreases to zero. 

By measuring the solubility, r, of the silver chloride in different concentration of added salt and extrapolating the solubilities to zero salt concentration, or better, to zero ionic strength, one obtains the solubility when v = 1. and from Eq. (29) K can be found. Then y can be calculated using this value of K and any measured solubility. Actually, this method is only applicable to sparingly soluble salts. Activity coefficients of ions and of electrolytes can be calculated from the Debye-HOckel equations. For a uni-univalent electrolyte, in water at 25 C, the equation for the activity coefficient of an electrolyte is... 

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 ... 

Table 2.3 References 1, Kirsten (1968) He, Ne, Ar values measured, Kr, Xe values by extrapolation according to equation (2.19) 2, Fisher (1970) 3, Hayatsu Waboso (1985) 4, Jambon et al. (1986) 5, Lux (1987) 6, Broadhurst et al. (1992) 7, Roselieb et al. (1992) 8, Shibata et al. (1998) 9, White et al. (1989) solubility values are normalized to 1300°C from the data given in White et al. (1995) by Carroll Stolper (1993) 10, Chamorro-Perez et al. (1996) 11, Carroll Stolper (1991) solubility value is calculated from the data given in Carroll Stolper (1991) for activity coefficient of Ar = 1 12 - Tables 4.1 and 4.3 and Equation (2.17).

The activity a2 of an electrolyte can be derived from the difference in behavior of real solutions and ideal solutions. For this purpose measurements are made of electromotive forces of cells, depression of freezing points, elevation of boiling points, solubility of electrolytes in mixed solutions and other characteristic properties of solutions. From the value of a2 thus determined the mean activity a+ is calculated using the equation (V-38) whereupon by application of the analytical concentration the activity coefficient is finally determined. The activity coefficients for sufficiently diluted solutions can also be calculated directly on the basis of the Debye-Hiickel theory, which will bo explained later on. 

For a more soluble (or completely miscible) pollutant (i.e., ethanol), a Henry s law constant can be similarly estimated by using its tabulated infinite dilution activity coefficient measured at Tx, by extrapolation using P2sat, from to T2 ... 

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