Regular solutions model, activity coefficients

The solubility of a gas is an integral part for the prediction of the permeation properties. Various models for the prediction of the solubility of gases in elastomeric polymers have been evaluated (57). Only a few models have been found to be suitable for predictive calculations. For this reason, a new model has been developed. This model is based on the entropic free volume activity coefficient model in combination with Hildebrand solubility parameters, which is commonly used for the theory of regular solutions. It has been demonstrated that mostly good results are obtained. An exception... 

Related Calculations. The regular-solution model of Scatchard and Hildebrand gives a fair representation of activity coefficients for many solutions containing nonpolar components. This procedure is suggested for estimating vapor-liquid equilibria if experimental data are not available. The solubility parameters and liquid molar volumes used as characteristic constants may be obtained from Table 1.10. For substances not listed there, the solubility parameters may be calculated from heat of vaporization and liquid molar volume data as shown in step 4. 

It was reported by Dixon and Johnston (56) that up to a moderate pressure, the liquid phase activity coefficient of the solid component 73 could be obtained by any activity coefficient model or, more conveniently, from the regular solution theory, as... 

The question of evaluating the liquid-phase activity coefficient of the solute species still remains. Although experimental data for y would be preferable, such data may not be available. Consequently, various liquid solution models and correlations are used. If the regular solution model is used, we have... 

The thermodynamic factor is evaluated for liquid mixtures from activity coefficient models. For a regular solution, for example,... 

The clay ion-exchange model assumes that the interactions of the various cations in any one clay type can be generalized and that the amount of exchange will be determined by the empirically determined cation-exchange capacity (CEC) of the clays in the injection zone. The aqueous-phase activity coefficients of the cations can be determined from a distribution-of-species code. The clay-phase activity coefficients are derived by assuming that the clay phase behaves as a regular solution and by applying conventional solution theory to the experimental equilibrium data in the literature.1 2 3... 

It is generally observed that as the temperature increases, real solutions tend to become more ideal and r can be interpreted as the temperature at which a regular solution becomes ideal. To give a physically meaningful representation of a system r should be a positive quantity and larger than the temperature of investigation. The activity coefficient of component A for various values of Q AB is shown as a function of temperature for t = 3000 K and xA = xB = 0.5 in Figure 9.3. The model approaches the ideal model as T - t. 

Recently, Rubingh ll) and Scamehorn et al. (9) have shown that the activity coefficients obtained by fitting the mixture CMC data can be correlated by assuming the mixed micelle to be a regular solution. This model proposed by Rubingh for binary mixtures has been extended to include multicomponent surfactant mixtures by Holland and Rubingh (10). Based on this concept Kamrath and Frances (11) have made extensive calculations for mixed micelle systems. 

Two models are frequently used to predict the activity coefficient of the solid the regular solution model (93) and the DLP (delta-lattice-parameter) model (94). With both models, the activity coefficient of component i, yf, is calculated in terms of the interaction parameter, ft, by the expression... 

The Chao-Seader and the Grayson-Streed methods are very similar in that they both use the same mathematical models for each phase. For the vapor, the Redlich-Kwong equation of state is used. This two-parameter generalized pressure-volume-temperature (P-V-T) expression is very convenient because only the critical constants of the mixture components are required for applications. For the liquid phase, both methods used the regular solution theory of Scatchard and Hildebrand (26) for the activity coefficient plus an empirical relationship for the reference liquid fugacity coefficient. Chao-Seader and Grayson-Streed derived different constants for these two liquid equations, however. 

Experimental vapor-liquid-equilibrium data for benzene(l)/n-heptane(2) system at 80°C (176°F) are given in Table 1.8. Calculate the vapor compositions in equilibrium with the corresponding liquid compositions, using the Scatchard-Hildebrand regular-solution model for the liquid-phase activity coefficient, and compare the calculated results with the experimentally determined composition. Ignore the nonideality in the vapor phase. Also calculate the solubility parameters for benzene and n-heptane using heat-of-vaporization data. 

The Scatchard-Hildebrand regular-solution model expresses the liquid activity coefficients y in a binary mixture as... 

The rational activity coefficients cannot be evaluated in any simple manner. Following the model of Truesdell and Christ (16), a regular solution approach to the problem can lead to expressions for the rational activity coefficients. If the exchange sites have the same charge and approximately the same size, then a symmetrical solid solution will be formed where the rational activity coefficients for the two components are given by ... 

Other molecular properties have been also proposed to model the hydrophobic interactions. The parachor, which is related to the surface tension of a compound (139, 140) represents mainly the intermolecular interactions in a liquid. The Hildebrand-Scott solubility parameter, 6, (141) is related to intermolecular van der Waals forces and the closely related molar attraction constant, F, is obtained by multiplying 6 by the molar volume (142). The partition coefficient between two solvents can be obtained from the solubility parameters and the molar volumes of the solute and the solvents (193). This relationship is based on regular solution theory (194) and the assumption that the partial molar volumes of the solute is not different from its molar volume. Recently this has been criticized and a new derivation was proposed (195) in which the partial molar volumes are taken into account. The molar refractivity, MR, is related to dispersion forces and can be obtained as a sum of the partial molar refractivi-ties assigned to atoms and bonds (140, 143). These parameters have been compared (144) to establish their relative applicability to correlations with biological activity. The conclusion was that logP and molecular refractivity were the best parameters. Parameters obtained from high pressure liquid chromatography (144,... 

A number of methods based on regular solution theory also are available. Only pure-component parameters are needed to make estimates, so they may be applied when UNIFAC group-interaction parameters are not available. The Hansen solubility parameter model divides the Hildebrand solubility parameter into three parts to obtain parameters 8d, 5p, and 5 accounting for nonpolar (dispersion), polar, and hydrogenbonding effects [Hansen,/. Paint Technot, 39, pp. 104-117 (1967)]) An activity coefficient may be estimated by using an equation of the form... 

For very dilute solutions, hence at low spreading pressiues, the concentrations of the adsorbates are also low, and the activity coefficients in the adsorbed phase approach unity. For real solutions, a suitable model of the activity coefficients must be used. Several such models have been suggested. The following equation, proposed by Gamba et al. [62] and based on the regular solution theory [63], was applied by Kaczmarski et al. [51] to account for the competitive isotherm of 1-indanol on cellulose tribenzoate ... 

In the liquid phase, the simplest option is an ideal liquid, with an activity coefficient equal to 1.0. That choice leads to Raoult s law, which may suffice for similar chemicals. Other models include regular solution theory using solubility parameters (although not in Aspen Plus), NRTL, Electrolyte NRTL, UNIFAC, UNIQUAC, Van Laar, and Wilson. Characteristics of the models are ... 

Thus, according to the model for a regular solution the activity coefficients Ya nd Yb are given by... 

Another possible extension is to consider an excess oil phase which is a mixtnre of two or more species. Provided that mixing within the micelle can still be considered ideal and that activity coefficients for all species in the bulk oil mixture are known, an expression for for each solnte is readily obtained. Micelles formed from surfactant mixtures can be treated provided that micelle composition is known or can be calculated from theories of mixed micelles such as regular solution theory and that solubilization is low enough not to affect micelle shape or composition. Finally, nonideal mixing in the micelles can be included if some model for the nonideality is available as well as data for evaluating the relevant parameters. Perhaps the simplest scheme for incorporating nonideality with nonpolar solutes is to use volume fractions instead of mole fractions in the spirit of Flory-Huggins theory. 

J. Sci.. in press). Indeed, in the case of a solid-solution with a small difference in the size of the substituting ions (relative to the size of the non-substituting ion), the first parameter, ao, is usually sufficient (8). Equations 5 and 6 then become identical to those of the "regular" solid-solution model of Hildebrand (9). For the case where both ao and ai parameters are needed, equations 5 and 6 become equivalent to those of the "subregular" solid-solution model of Thompson and Waldbaum (10). a model much used in high-temperature work. Equations 5 and 6 can also be shown equivalent to Margules activity coefficient series (11). 

Repeat the calculations of the previous problem with the regular solution model. Compare the two results. Develop an expression for the activity coefficient of a species in a mixture from the Peng-Robinson equation of state with the van der Waals one-fluid mixing rules, a. Show that the minimum amount of work, W , necessary to separate 1 mole of a binary mixture into its pure components at constant temperature and pressure is... 

Use the regular solution model to predict the activity coefficients of benzene and 2,2.4-trimethyl pentane in their mi.xtures at 55 C. What are the predicted values of the infinite-dilution activity coefficients ... 

In these equations the vapor compositions, vb and yc. and the equilibrium pressure P are unknown (the equilibrium pressure is 1.013 bar only at xb — 0.525). The solution is obtained by choosing a value of xg, using xc = 1 — xb, and copiputing ys and yc from Eqs. i, and the total pressure from Eq. iii. The vapor-phase mole fractions are then computed from Eqs. ii. The results of this calculation are given in the table and Fig. 2. Regular solution model. Since benzene and cyclohexane are nonpolar, and their solubility parameters are given in Table 9.6-1, the activity coefficients can be predicted using Eqs. 

If.. separately, we knew the activity coefficients for bromine in carbon tetrachloride, we could u.se the data in the table to evaluate the activity coefficient of bromine in water (Problem 11.4-3). Since the regular solution model can be used to represent the Bri-CC mixture, we can then surmise that will be on the order of magnitude of unity. This suggests that y will be a large number, on the order of 100 or more. Such behavior for the activity coefficient o f the minor component is not unusual in mixtures of species with such different molecular characteristics as strongly quadrupolar liquid bromine and strongly polar and hydrogen-bonded water. H... 

If the liquid.mixture is ideal, so that y = 1, we have the case of ideal solubility of a solid in a liquid, and the solubility can be computed from only thermodynamic data and ACp) for the solid species near the melting point. For nonideal solutions, yi must be estimated from either experimental data or a liquid solution model, for example, UNIFAC. Alternatively, the regular solution theory estimate for this activity coefficient is... 

Although the discussion of this section has centered on the solubility of a solid in a pure liquid, the methods used can be easily extended to mixed solvents. In fact, to apply the equations developed in this section to mixed solvents, we need only recognize that the measured or computed value of 71, the activity coefficient for the dissolved solid, used in the calculations must be appropriate to the solid and mixed solvent combination being considered. In the regular solution model, for example, this means replacing Eq. 

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