Binary activity coefficients, regular

In Chapter 4, methods based on equations of state were presented for predicting thermodynamic properties of vapor and liquid mixtures. Alternatively, as developed in this chapter, predictions of liquid properties can be based on correlations for liquid-phase activity coefficients. Regular solution theory, which can be applied to mixtures of nonpolar compounds using only properties of the pure components, is the first type of correlation presented. This presentation is followed by a discussion of several correlations that can be applied to mixtures containing polar compounds, provided that experimental data are available to determine the binary interaction parameters contained in the correlations. If not, group-contribution methods, which have recently undergone extensive development, can be used to make estimates. All the correlations discussed can be applied to predict vapor-liquid phase equilibria and some, as discussed in the final section of this chapter, can estimate liquid-liquid equilibria. 

But for binary systems only one of the above two equations is independent, as by summing these two equations we get zero in the RHS and also zero in the LHS. The LHS equalling to zero is due to the Gibbs-Duhem relation. To illustrate this, we take the following equations for the activity coefficients (regular solution model)... 

If the mutual solubilities of the solvents A and B are small, and the systems are dilute in C, the ratio ni can be estimated from the activity coefficients at infinite dilution. The infinite dilution activity coefficients of many organic systems have been correlated in terms of stmctural contributions (24), a method recommended by others (5). In the more general case of nondilute systems where there is significant mutual solubiUty between the two solvents, regular solution theory must be appHed. Several methods of correlation and prediction have been reviewed (23). The universal quasichemical (UNIQUAC) equation has been recommended (25), which uses binary parameters to predict multicomponent equihbria (see Eengineering, chemical DATA correlation). 

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... 

Consideration of the thermodynamics of nonideal mixing provides a way to determine the appropriate form for the activity coefficients and establish a relationship between the measured enthalpies of mixing and the regular solution approximation. For example, the excess free energy of mixing for a binary mixture can be written as... 

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. 

By analogy with the treatment of mixed micelles, we now assume that the free energy of mixing of the surface phase can be calculated using the standard regular solution expression for the activity coefficients in a binary mixture ... 

By extending regular solution theory for binary mixtures of AEg in aqueous solution to the adsorption of mixture components on the surface (3,4), it is possible to calculate the mole fraction of AEg, Xg, on the mixed surface layer at tt=20, the molecular interaction parameter, 6, the activity coefficients of AEg on the mixed surface layer, fqg and f2s and mole concentration of surfactant solution, CTf=20 3t surface pressure tt=20 mn-m l (254p.l°C). The results from the following equations are shown in Table I and Table II. 

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

What are the equations for computing the Gibbs free energy, enthalpy, and entropy of formation of a binary symmetrical regular solution How are the rational activity coefficients (/I values for the solid components) related to their mole fractions in such a solid solution ... 

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... 

We next turn our attention to a consideration of some of the properties of regular solutions that differ from ideal solutions. We consider a binary solution. In this case, the Margules formalism for the expansion of the activity coefficient in powers of the mole fraction (Sec. 11-6) and use of Eq. (11-119) give... 

Perhaps the most important term in Eq. (5.2-3) is the liquid-phase activity coefficient, and methods for its prediction have been developed in maiiy forms and by many workers. For binary systems the Van Laar (Eq. (1.4-18)], Wilson [Eq. (1.4-23)], NRTL (Eq. (1.4-27)], and UmQUAC [Eq. (t.4-3ti)] relationships are useful for predicting liqnid-iffiase nonidealities, but they require some experimental data. When no dim are available, and an approximate nonideality correction will suffice, the UNIFAC approach (Eq. (1.4-31)], which utilizes functional group contributions, may be used. For special cases involving regular solutions (no excess entropy of mixing), the Scatchard-Hildebrand method provides liquid-phase activity coefficients based on easily obtained pure-component properties. 

The regular solution theory is not limited to binary systems. It can directly be applied for the calculation of activity coefficients in multicomponent systems ... 

The rest of this chapter is in two parts. First, we consider how to calculate mole fractions in solid solutions assumed to have only long range order, and how to combine these mole fractions into Raoultian activities. Then we consider the determination of activity coefficients in (binary) solid solutions, and how regular solution and Margules equations are used to systematize these. 

The term general solution was introduced by Flory to characterize polymer solutions whose enthalpy of mixing is not zero. The model of general solutions borrows the formula of excess enthalpy from regular systems and the excess entropy from athermal solutions. Thus, a treatment of non-ideal polymer solutions arises which is simpler than the conventional methods applied to real systems this allows the deduction, on the basis of the known relationships, of the expressions of functions of deviation from ideality. Thus, for the activity coefficients of components in a binary system the following relations were established ... 

For the present purposes the term regular binary solution may be taken as meaning a solution having activity coefficients given by the expressions quoted. Dividing the first of these expressions by the second, we obtain... 

If above relatiOTi can not be determined, it is sometimes possible to deduce activity coefficient values for components in solid phase based on thermodynamic solid solutimi model. For example, if symmetrical solvus (Fig. 1.2) exists for a binary system, regular solution model could be applicable to the estimation of activity coefficients and other thermodynamic parameters values of solid solution... 

To a first approximation, the deviations from ideality may be expressed by regular solution theory (Guggenheim, 1952, p. 29), in which the activity coefficient of a component in a binary system is given by the simple expression ... 

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