Activity coefficient equations symmetrical

You may have noticed that the Margules model is equivalent to the two-parameter Redlich-Kister polynomial used in Examples 12.6 and 12.7. This maybe confirmed by setting A12 = Oo andA = Oo + Oi to the above equations (see Example 12.8. below). In the form given here, the expressions for the activity coefficients are symmetric in the two components such that each expression is obtained from each other by switching the subscripts 1 and 2. 

We see that for the symmetrical equation (and most other useful activity coefficient equations) the activity coefficient of the solute is proportional to the square of the concentration of the solvent. For most gases dissolved in liquids, the change in solute concentrations is so small that the concentration of the solvent is practically constant, 1.00. Thus, over the range of practical interest the activity coefficient— Raoult s law type—of the dissolved solute gas is practically constant, which is the same as saying that Henry s law is obeyed within experimental accuracy, with. extrapolated iT-... 

Most widely used liquid-phase activity coefficient equations represent the group g liJiTXaXj as some relatively simple algebraic function of the hquid mol fractions. If we choose g l(RTXaXi,) = some constant, we find the symmetrical equation, which is the simplest activity coefficient equation which is consistent with the Gibbs-Duhem equation. More complex functions are more successful at fitting experimental VLE data. 

FIGURE 11.7 Calculated values of the molar Gibbs energy of a binary mixture at 25°C, assuming the symmetrical activity coefficient equation, with various values of A. 

For such components, as the composition of the solution approaches that of the pure liquid, the fugacity becomes equal to the mole fraction multiplied by the standard-state fugacity. In this case,the standard-state fugacity for component i is the fugacity of pure liquid i at system temperature T. In many cases all the components in a liquid mixture are condensable and Equation (13) is therefore used for all components in this case, since all components are treated alike, the normalization of activity coefficients is said to follow the symmetric convention. ... 

Equation (16) is a differential equation and applies equally to activity coefficients normalized by the symmetric or unsymme-tric convention. It is only in the integrated form of the Gibbs-Duhem equation that the type of normalization enters as a boundary condition. 

To overcome this difficulty the Debye-Hiickel theory was expanded for symmetrical valence-type electrolytes, and the complex functions in the expansion (1/2 X3 — 2Y3) and (1/2 X5 — 4Y5) calculated and published (26). The result of these expansions is to add a term Eext to the equation for the activity coefficient given in Equation 2. For symmetric valence-type electrolytes such as hydrobromic acid this term is... 

Because the activities of species in the exchanger phase are not well defined in equation 2, a simplified model—that of an ideal mixture—is usually employed to calculate these activities according to the approach introduced bv Vanselow (20). Because of the approximate nature of this assumption and the fact that the mechanisms involved in ion exchange are influenced by factors (such as specific sorption) not represented by an ideal mixture, ion-exchange constants are strongly dependent on solution- and solid-phase characteristics. Thus, they are actually conditional equilibrium constants, more commonly referred to as selectivity coefficients. Both mole and equivalent fractions of cations have been used to represent the activities of species in the exchanger phase. Townsend (21) demonstrated that both the mole and equivalent fraction conventions are thermodynamically valid and that their use leads to solid-phase activity coefficients that differ but are entirely symmetrical and complementary. 

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 ... 

Using these equations and the expression already deduced for (Equations 10.36 and 10.39) the Debye-Hiickel equation for the mean activity coefficient of a symmetrical electrolytes such as NaCl is ... 

Two interesting features of the one-constant Margules equations are apparent from this figure. First, the two species activity coefficients are mirror images of each other as a function of the composition. This is not a general result, but follows from the choice of a symmetric function in the compositions for G . Second, yi 1 as Xj —> 1,... 

The one-constant Margules equation provides a satisfactory representation for activity coefficient behavior only for liquid mixtures containing constituents of similar size, shape, and chemical nature. For more complicated systems, particularly mixtures of dissimilar molecules, simple relations such as Eq. 9.5-1 or 9.5-5 are not valid. In particular, the excess Gibbs energy of a general mixture will not be a symmetric function of the mole fraction, and the activity coefficients of the two species in a mixture should not be expected to be mirror images. One possible generalization of Eq. 9.5-1 to such cases is to set... 

In this equation i denotes the species and has values of 1 and 2. These results are known as the two-constant Margules equations. In this case the excess Gibbs energy is not symmetric in the mole fractions and the two activity coefficients are not mirror images of each other as a function of concentration. 

The plots indicate that the non-ideality is significant at higher concentrations. Consequently, liquid activity coefficients should be used. Because NHj is subcritical, there is a choice between symmetric and asymmetric conventions. In the first case, Antoine equation is necessary for both components. In the second case, Henry constant for ammonia-water pair is required, but this may be obtained from the slope of the experimental curves plotted in Fig. 6.14. Vapour pressures and Henry constants are ... 

The constants in these equations are those obtained from the end values of the activity coefficients of the three binary systems, each of which is symmetrical ... 

FIGURE 9.4 Comparison of the acetone-water liquid-phase activity coefficients computed from the VLE data (copied from Figure 8.6) and their representation by the symmetrical and Margules equations. 

For liquids and solids, determine the activity coefficients for binary and multicomponent mixtures through activity coefficient models, including the two-suffix Margules equation, the three-suffix Margules equation, the van Laar equation, and the Wilson equation. Identify when the symmetric activity coefficient model is appropriate and when you need to use an asymmetric model. 

Equation (E7.9F) shows that if the polarizabilities are equal, we have an ideal solution in all other cases A > 0 (for spherically symmetric nonpolar molecules) Thus, we are much more likely to find in nature a case where like interactions dominate (i.e., have lower energy) and the activity coefficients (based on the Lewis/RandaU reference state) are greater than 1 for example, see Equations (7.55) and (7.56). 

Examination of Equation (7.57) [or Equations (7.55) and (7.56)] already shows one limitation in our model for g . It is completely symmetric that is, if we interchanged a and h in this equation, it would be no different. Accordingly, it could not be used to model systems in which the activity coefficients are not symmetric, such as that depicted in Example 7.7. Such asymmetric activity coefficient models will be presented in the next section. 

Since the value for A based on species a is close to that from species b, the system is reasonably symmetric and we can use the two-suffix Margules equation. Alternatively, we could use an asymmetric activity coefficient model the van Laar equation is commonly used for azeotropes. 

To get the main idea of the charge effect on adsorption kinetics, it is sufficient to consider an aqueous solution of a symmetric (z z) ionic surfactant in the presence of an additional indifferent symmetric (z z) electrolyte. When a new interface is created or the equilibrium state of an interfacial layer disturbed a diffusion transport of surface active ions, counterions and coions sets in. This transport is affected by the electric field in the DEL. According to Borwankar and Wasan [102], the Gouy plane as the dividing surface marks the boundary between the diffuse and Stem layers (see Fig. 4.10). When we denote the surfactant ion, the counterion and the coion, respectively, with the indices / = 1, 2 and 3, the transport of the ionic species with valency Z/ and diffusion coefficient A, under the influence of electrical potential i, is described by the equation [2, 33] ... 

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