Thermodynamic Factors for Binary Systems

The symbol X is used to indicate that the differentiation of Inyi with respect to mole fraction Xj is to be carried out subject to the restriction that + X2 = 1. For a regular solution, for example, the activity coefficient of component 1 is given by 

If we replace the mole fraction X2 with 1 - x and differentiate with respect to x we find d In 7i 

Few real systems are adequately represented by the regular solution model. As a result, there are several more complicated models of G that have been proposed. For these 

The activity coefficients 7 and may be expressed in terms of Q and its composition derivatives by (see Section D.2 for derivation) 

That is, Qi is the partial derivative of Q with respect to mole fraction Xj, while holding mole fraction X2 constant, and Q2 is the partial derivative of Q with respect to mole fraction X2, while holding mole fraction Xj constant. 

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Diffusion coefficients in binary liquid mixtures can be strong functions of composition. To illustrate this fact we have plotted experimental data for a few systems in Figure 4.1. The Maxwell-Stefan coefficient > also is shown in Figure 4.1. To obtain the Maxwell-Stefan coefficients we have divided the Fick D by the thermodynamic factor F... 

Rathbun and Babb (1966) used Eq. 4.1.9 for D° and found that a value of m = 0.6 gave a good fit to data for a few systems that exhibited positive deviations from Raoult s law, whereas m — 0.3 worked well for systems having negative deviations from Raoult s law. Kosanovich and Cullinan (1976) found that an exponent of 0.5 on the thermodynamic factor reproduced the concentration dependence of several nonideal binary mixtures quite nicely. 

Sanni and Hutchison (1973) presented data on the binary Fick dilfusivity for the systems benzene-chloroform, cyclohexane-carbon tetrachloride, cyclohexane-toluene, benzene-cyclohexane, benzene-toluene, and diethyl ether-chloroform. Calculate the thermodynamic factor F for these systems using parameters from Gmehling and Onken (1977ff). Hence, estimate the Maxwell-Stefan diffusion coefficients and test the applicability of the Vignes model. 

If the K factors of a binary system are known as a function of pressure and temperature, the entire phase diagram can be constructed. As we will see in Chapter lo. thermodynamics provides methodologies for the estimation of these factors. 

Azeotropic behavior is not limited to binary systems only. Also ternary and quaternary azeotropic points are observed. For the determination of the azeotropic points in ternary and quaternary systems, thermodynamic models (g -models, equations of state, group contribution methods) can again be applied [40]. Azeotropic points in homogeneous systems can be found with the help of nonlinear regression methods. At the azeotropic point all separation factors aij show a value of 1 in the case of homogeneous systems. This means that the following condition has to be fulfilled ... 

In most commercial process simulators, model parameters for pure component properties and binary parameters can be found for a large number of compounds and binary systems. However, the simulator providers repeatedly warn in their software documentations and user manuals that these default parameters should be applied only after careful examination by the company s thermodynamic experts prior to process simulation. For verification of the model parameters again, a large factual data bank like the DDE is the ideal tool. The DDE allows checking all the parameters used for the description of the pure component properties as a function of temperature and of the binary parameters of a multicomponent system by access to the experimental data stored. On the basis of the results for the different pure component properties and phase equilibria, excess enthalpies, activity coefficients at infinite dilution, separation factors, and so on, the experienced chemical engineer can decide whether all the data and parameters are sufficiently reliable for process simulation. 

For monodisperse primary chains, we have a strictly two-component system, and the thermodynamic stability limit (spinodal) is given by a cj), T) = 0, where cr is the factor (7.123). Further, for such strictly binary systems, the critical solution point, if it exists in the pregel regime, can be found by the additional condition d A o/dcp = 0. The condition is given explicitly by... 

The thermodynamic basis to explain miscibility in polymer blends is an exothermic heat of mixing as entropic contributions are small for such systems. Intramolecular repulsions may be an important factor in realizing exothermic heat of mixing. The application of binary interaction model to predict a compositional window of miscibility for copolymer/homopolymer blends, terpolymer blends with common monomers, copolymer blends with common monomers, is illustrated. A 6 X parameter expression for free energy of mixing for two copolymers with four monomers is described. The spinodal is derived from stability and miscibility considerations. The physical meaning behind concave and convex curvature of phase envelopes is described. 

WhethCT the formation of a one-phase binary amorphous system is thermodynamically favorable is determined by different factors. In any case, for a system to be thermodynamically miscible, the free energy, given by Eq. 2.33, must be negative ... 

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