Energy of binary mixing

Interactions between species can be either attractive or repulsive. In most experimental situations, mixing occurs at constant pressure and the enthalpic interactions between species must be analysed to find a minimum 

Regular solution theory writes the energy of mixing in terms of three pairwise interaction energies (u a, ubb) between adjacent 

The corresponSing energy of a B-monomer with one of its neighbours is similar to Eq. (4.14)  

Each lattice site of a regular lattice has z nearest neighbours, where z is 

Denoting the volume fraction of species A by = a = 1 (ps, Eqs (4.14)-(4.16) are combined to get the total interaction energy of a binary mixture with n lattice sites  

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The expression for the total free energy of binary mixing corresponding to equations (5) and (18) is ... 

Another approach has been intensively developed recently by Puvvada and Blankschtein, who considered also the formation of mixed micelles in surfactant mixtures [20-22]. For example, the free energy of formation of binary mixed surfactant micelle, containing Ha molecules of the surfactant A, ns molecules of the surfactant B and, n water moleeules in thermodynamic equilibrium was expressed as... 

The most relevant theory for modeling the free energy of binary polymer mixtures is the Flory-Huggins theory, initially employed for solvent-solvent and polymer-solvent mixtures. This theory was independently derived by Flory [4, 5] and Huggins [6, 7]. The key equation (combined from discussions earlier in this chapter on entropy and enthalpy of mixing) is ... 

Figure C2.1.10. (a) Gibbs energy of mixing as a function of the volume fraction of polymer A for a symmetric binary polymer mixture = Ag = N. The curves are obtained from equation (C2.1.9 ). (b) Phase diagram of a symmetric polymer mixture = Ag = A. The full curve is the binodal and delimits the homogeneous region from that of the two-phase stmcture. The broken curve is the spinodal.

Physical Equilibria and Solvent Selection. In order for two separate Hquid phases to exist in equiHbrium, there must be a considerable degree of thermodynamically nonideal behavior. If the Gibbs free energy, G, of a mixture of two solutions exceeds the energies of the initial solutions, mixing does not occur and the system remains in two phases. Eor the binary system containing only components A and B, the condition (22) for the formation of two phases is... 

The physical reason for the inherent lack of incentive for mixing in a polymer-polymer system is related to that already cited in explanation of the dissymmetry of the phase diagram for a polymer-solvent binary system. The entropy to be gained by intermixing of the polymer molecules is very small owing to the small numbers of molecules involved. Hence an almost trivial positive free energy of interaction suffices to counteract this small entropy of mixing. 

In the binary solution considered, the free energy of the system before mixing is the sum of the free energies of the pure components ... 

The equality of aA and PA/PA has been explained earlier. It may be recalled that the activity of the component A in the solution is defined by this equality. For the binary solution A-B, the integral molar free energy of mixing is then... 

Fig. 1. Variation of free energy of mixing with composition for a binary alloy system possessing a miscibility gap in the range X-Z.

It is also of interest to note that starting from miscibility gap data, attempts have been made to derive thermodynamic properties, i.e., free energies of mixing, and Sundquist (23) summarizes information and discusses a number of binary alloy systems. 

Once it has been established that the components of a binary monolayer are to some degree miscible, the energetics of their interaction may be calculated directly from the 11/A isotherms of the mixture and its individual components. As proposed by Goodrich (1957), this technique employs the differences in the work of compression of the binary film and the work required to compress each of the films of the pure components to the same surface pressure. The result is the total free energy of mixing as expressed by the sum of the excess and ideal free energies of mixing in (14), where Nt... 

Binary solutions have been extensively studied in the last century and a whole range of different analytical models for the molar Gibbs energy of mixing have evolved in the literature. Some of these expressions are based on statistical mechanics, as we will show in Chapter 9. However, in situations where the intention is to find mathematical expressions that are easy to handle, that reproduce experimental data and that are easily incorporated in computations, polynomial expressions obviously have an advantage. 

A binary ionic solution must contain at least three kinds of species. One example is a solution of AC and BC. Here we have two cation species A+ and B+ and one common anion species C . The sum of the charge of the cations and the anions must be equal to satisfy electro-neutrality. Hence NA+ + NB+ = N(. = N where NA+, AB+ and Nc are the total number of each of the ions and N is the total number of sites in each sub-lattice. The total number of distinguishable arrangements of A+ and B+ cations on the cation sub-lattice is M/N A, JVg+ . The expression for the molar Gibbs energy of mixing of the ideal ionic solution AC-BC is thus analogous to that derived in Section 9.1 and can be expressed as... 

Thus, in the free energy of mixing of a binary system, the first-order terms cancel each other and do not appear. All of the integrals contained in the terms Z a, A, K, M, and T in Eq. (87) are dependent solely on the properties of the comparison salt and are constant for binary conformal ionic mixtures having X- as the anion. 

Another type of ternary electrolyte system consists of two solvents and one salt, such as methanol-water-NaBr. Vapor-liquid equilibrium of such mixed solvent electrolyte systems has never been studied with a thermodynamic model that takes into account the presence of salts explicitly. However, it should be recognized that the interaction parameters of solvent-salt binary systems are functions of the mixed solvent dielectric constant since the ion-molecular electrostatic interaction energies, gma and gmc, depend on the reciprocal of the dielectric constant of the solvent (Robinson and Stokes, (13)). Pure component parameters, such as gmm and gca, are not functions of dielectric constant. Results of data correlation on vapor-liquid equilibrium of methanol-water-NaBr and methanol-water-LiCl at 298.15°K are shown in Tables 9 and 10. 

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... 

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