Activity coefficient from experimental data

Mean Ionic Activity Coefficients from Experimental Data... 

Setting Yi = 1 (ideal solution) this reverts to Raoult s law. We may view then the activity coefficient as a correction to Raoult s law that accounts for deviations from ideal-solution behavior. Equation (12.24) is the basis of VLE calculations, provided that pressure is sufficiently low. It is also used to extract activity coefficients from experimental data, as we will see with examples below. 

Use of the Meissner family of curves as presented in Chapter IV, Figure (4.6), in order to correlate the reduced activity coefficients of strong electrolytes. Meissner found that the reduced activity coefficients of strong electrolytes fell into a pattern which he formalized. If it can be assumed that the reduced activity coefficients of strong electrolytes follow this pattern, then deviation from these curves may be said to indicate that the species in question does not completely dissociate or may be forming complexes. If activity coefficients from experimental data are available they may be plotted on the Meissner chart. Severe deviation from the curves, such as the crossing of lines, may be taken to indicate complex formation. However, there would be no indication of what the complexes might be. 

This simple equation is adequate to our present purpose, allowing easy calculation of activity coefficients from experimental low-pressure VLE data. For comparison, when a system obeys Raoult s law, y(P = P at, and yf = 1. 

Compute logarithms of the activity coefficients from experimental X-Y data. Assuming an ideal vapor phase, the activity coefficients are given as y, = P 7, / X, P( > and y2 = PY2/X2 P2. The natural logarithms of the activity coefficients calculated using the preceding equations are shown in cols. 7 and 8 of Table 1.11. 

The development of equations that successfully predict multicomponent phase equilibrium data from binary data with remarkable accuracy for engineering purposes not only improves the accuracy of tray-to-tray calculations but also lessens the amount of experimentation required to establish the phase equilibrium data. Such equations are the Wilson equation (13), the non-random two-liquid (NRTL) equation (14), and the local effective mole fractions (LEMF) equation (15, 16), a two-parameter version of the basically three-parameter NRTL equation. Larson and Tassios (17) showed that the Wilson and NRTL equations predict accurately ternary activity coefficients from binary data Hankin-son et al. (18) demonstrated that the Wilson equation predicts accurately... 

An alternative approach is to estimate activity coefficients of the solvents from experimental data and correlate these coefficients using, for example, the Wilson equation. Rousseau et al. (3) and Jaques and Furter (4) have used the Wilson equation, as well as other integrated forms of the Gibbs-Duhem equation, to show the utility of this approach. These authors found it necessary, however, to modify the definitions of the solvent reference states so that the results could be normalized. 

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... 

A general formulation of the problem of solid-liquid phase equilibrium in quaternary systems was presented and required the evaluation of two thermodynamic quantities, By and Ty. Four methods for calculating Gy from experimental data were suggested. With these methods, reliable values of Gy for most compound semiconductors could be determined. The term Ty involves the deviation of the liquid solution from ideal behavior relative to that in the solid. This term is less important than the individual activity coefficients because of a partial cancellation of the composition and temperature dependence of the individual activity coefficients. The thermodynamic data base available for liquid mixtures is far more extensive than that for solid solutions. Future work aimed at measurement of solid-mixture properties would be helpful in identifying miscibility limits and their relation to LPE as a problem of constrained equilibrium. 

From the above equations it is evident that in any calculation of activity coefficients ionic strength is a key parameter and as such it is useful if it can be calculated from experimental data. Ionic strength can be calculated from electrical conductivity measurements using the Babcock equation which is given by Sposito (1989) as... 

All current activity coefficient estimation models are by necessity semi-empirical in nature, because too little is known about solution theory for outright estimation. Chemical modeling is not readily available and is not far enough developed to make this type of calculation. The constants required by these models must be estimated using either experimental data (e.g. an infinite dilution activity coefficient or a molar volume) or group contributions derived from experimental data (e.g. interaction constants, molecular volumes and surface areas). 

The only remaining undetermined thermodynamic properties are yH 0 and yEtQn Because or the highly nonideal behavior of a liquid solution of ethanol and wat these must be determined from experimental data. The required data, found fro VLE measurements, are given by Otsuki and Williams.t From their results for t ethanol/water system one can estimate values of yH2G and yEt0H at 200°C, (Pressu has little effect on the activity coefficients of liquids.)... 

The solubility product is equal to Ca x AI(OH)4 x [OH , where curly brackets denote species activities and [OH ] may be replaced by 2[Ca ] - [A1(0H)4"]. As the concentrations are low, activity coefficients may be calculated from simplified Debye-Hiickel theory. Solubility products may thus be obtained from experimental data (NI8,B118,BI 19.C48) (Table 10.2). The variations in solubility products with temperature may be represented by empirical equations of the form... 

Although Procedure C is a good predictive method, it should not be used as a substitute to reducing good experimental data to obtain activity coefficients. In general, higher accuracy can be obtained from empirical models when these models are used with binary interaction parameters obtained from experimental data. 

This method is to be used to estimate the activity coefficient of a low molecular weight solvent in a solution with a polymer. This procedure, unlike the other procedures in this chapter, is a correlation method because it requires the Flory-Huggins interaction parameter for the polymer-solvent pair which must be obtained from an independent tabulation or regressed from experimental data. In addition, the specific volumes and the molecular weights of the pure solvent and the pure polymer are needed. The number average molecular weight of the polymer is recommended. The method cannot be used to estimate the activity of the polymer in the solution. 

In another attempt (Fawcett and Tikenen, 1996), the introduction of a changing dielectric constant of the solvent (although taken from experimental data) as a function of concentration has been used to estimate activity coefficients of simple 1 1 electrolyte solutions for concentrations up to 2.5 mol dm". ... 

System number Cosolvent Solute Reference Deviation from experimental data Flory-Huggins Wilson activity activity coefficients coefficients ... 

It should be emphasized that the parameters involved in the activity coefficients are adjustable parameters which cannot be obtained easily from the properties of the mixed solvents, for instance the vapor-liquid equilibria. However, for the solubilities of structurally related caffeine and theophyllene in water/iV,iV-dimethylformamide, the values of the Wilson parameters are close to each other (1.96 and 0.12 for caffeine and 1.81 and 0.10 for theophyllene). If the Wilson parameters for theophyllene are used to predict the solubility of caffeine in water/iV,A -dimethylformamide, a deviation of 8.8% from experimental data is obtained. The deviation was, however, 6.5% when the Wilson parameters were determined by fitting the experimental solubility data (Table 1). The values of the Wilson parameters determined from the solubilities of the structurally more different sulfonamides (sulfadiazine, sulfadimidine, sulfamethizole, sulfamethoxazole, sulfapyridine, sulfamethoxypyridazine, sulfanilamide and sulfisomi-dine) in water/dioxane mixtures are listed in Table 3. Even for such cases, the average values of the Wilson parameters can be used for a first estimation of the solubilities of the above group of drugs (Table 3). 

We focus on the thermodynamic models that deal with the liquid mixtures in this chapter. From the two categories of activity coefficient models, the correlative one is not very useful for solubility prediction and solvent screening purposes. The main reason for this is the lack of experimental data for the binary interaction parameters of the solute-solvent, solute-antisolvent, and solvent-antisolvent systems. As an example, the activity coefficient from... 

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