Activity coefficients regular solution

Scatchard Hlldebrand regular-solution activity coefficients. Hildebrand (1929) defined a regular solution as the mixture in which components mix with no excess entropy provided there is no volume change on mixing. Scatchard in an independent work arrived at the same conclusion. The definition of regular solutions (Hildebrand and Scott, 1950) is in line with van Laar s assumption that the excess entropy and the excess volume of mixing are negligible. Scatchard and Hildebrand used an approach different from van Laar s to calculate G. They defined parameter C as... 

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... 

We will now have a look at the standard states for each of these cases. These states are very different from each other, and some are very hypothetical and seemingly very unrealistic. We will try to show how this arises, and that the standard states we use are actually quite reasonable. To do this with real numbers rather than just symbols, we need either experimental data or some equations that simulate or fit experimental data in a realistic way. Real data unfortunately have uncertainties, so we will borrow the concept of regular solutions from Chapter 10. For our purposes here, all we need to know is that in such solutions, activity coefficients follow the relationships... 

Hydrocarbon mixtures can be assumed to be regular solutions it is thus possible to estimate the activity coefficient using a relation published by Hildebrandt (1950) ... 

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). 

P7.2 In a regular solution, the activity coefficients are given by the equations... 

The behaviour of most metallurgically important solutions could be described by certain simple laws. These laws and several other pertinent aspects of solution behaviour are described in this section. The laws of Raoult, Henry and Sievert are presented first. Next, certain parameters such as activity, activity coefficient, chemical potential, and relative partial and integral molar free energies, which are essential for thermodynamic detailing of solution behaviour, are defined. This is followed by a discussion on the Gibbs-Duhem equation and ideal and nonideal solutions. The special case of nonideal solutions, termed as a regular solution, is then presented wherein the concept of excess thermodynamic functions has been used. 

In relation to the separation of cadmium from zinc by volatilization, it is worth noting that the ratio Pcd/P2n increases with decreasing temperature (from 5.09 at 850 °C to 7.30 at 650 °C and 12.69 at 450 °C). The liquid solution of zinc and cadmium exhibits a regular solution behavior and, therefore, the activity coefficient of cadmium dissolved in liquid zinc increases with decreasing temperature. Both these features suggest that the elimination of cadmium from zinc is more efficient at relatively lower temperatures. This is reflected in the choice of the temperatures in the second column. 

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... 

These expressions can be simplified since the activity coefficient in the particular case of a regular solution can be expressed by the regular solution constant 2 through eqs. (3.84) and (3.85) ... 

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. 

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 ... 

Introducing the regular solution expression for the activity coefficients (Equations 14 and 15 with superscript s replaced by m) into Equation 24 Rubingh found the following useful relations ... 

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 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... 

In terms of activity coefficients, the regular-solution approximation can also be expressed as RTIn y% = RTIn yi = iVhx. ) The mixing parameter Cl expresses nonideality... 

Extrapolation of Kc to infinite dilution to give K is usually easy because the activity coefficients of most ionic substances vary in a regular manner with ionic strength and follow the Debye-Hiickel equation (Eq. 6-33) in very dilute solutions (ionic strength < 0.01). 

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. 

According to a theory, based on the regular solution theory, a deviation from ideal behaviour can be described by the introduction of the activity coefficients / and f2-... 

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