Energy Gibbs’ free, of mixing

Figure A2.5.15. The molar Gibbs free energy of mixing versus mole fraetionxfor a simple mixture at several temperatures. Beeause of the synuuetry of equation (A2.5.15) the tangent lines indieating two-phase equilibrium are horizontal. The dashed and dotted eiirves have the same signifieanee as in previous figures.

Fig. 25. Relationship between the measured interfacial strength and the (negative) Gibbs free energy of mixing, (-AG )o5, for glass beads treated with various silane coupling agents embedded in a PVB matrix. Error bars correspond to 95% mean confidence intervals. Redrawn from ref. [165].

To obtain AmixGm, the molar Gibbs free energy of mixing, we divide equation (7.5) by n — J2 n> t0 obtain... 

V/1 is the molar volume of the solvent, ns and np the molar concentrations of the solvent and the polymer, respectively, and AGm the Gibbs free energy of mixing. Equation (26) reduces in the limit of infinite dilution to the well-known Van t Hoff equation... 

By simple thermodynamic arguments Brown14 has shown that, consistent with the accuracy of this second-order approximation, one may obtain from the form of Eq. (87) the form of the excess Gibbs free energy of mixing (AG ), the enthalpy of mixing of a molten salt (AHm), and the deviation of the surface tension from linearity ... 

Gibbs free energy of mixing per unit volume... 

Interaction parameter does depend on T and P, and the excess Gibbs free energy of mixing is described as in the preceding model ... 

If Aq = A = A2 = 0, the excess Gibbs free energy of mixing is zero throughout the compositional field and the mixture is idea/. 

Figure 3.9D shows the form of the curve of the excess Gibbs free energy of mixing obtained with Van Laar parameters variable with T. the mixture is subregular— i.e., asymmetric over the binary compositional field. 

Because the ideal Gibbs free energy of mixing contribution is readily generalized to n-component systems (cf eq. 3.131), the discussion involves only excess terms. 

According to the Hillert model (Hillert, 1980), the excess Gibbs free energy of mixing of a ternary mixture is given by... 

Let us now imagine that we are dealing with a regular mixture (A,B)N with an interaction parameter W = +20 kJ/mole. The Gibbs free energy of mixing at various temperatures will be... 

The Gibbs free energy of mixing curves will have the form shown in figure 3.10A. By application of the above principles valid at equilibrium conditions, we deduce that the minimum Gibbs free energy of the system, at low T, will be... 

Figure 3,10 Solvus and spinodal decomposition fields in regular (B) and subregular (D) mixtures. Gibbs free energy of mixing curves are plotted at various T conditions in upper part of figure (A and C, respectively). The critical temperature of unmixing (or consolute temperature ) is the highest T at which unmixing takes place and, in a regular mixture (B), is reached at the point of symmetry.

Figure 3.10 also shows the fields of spinodal decomposition defined by the loci of the points of inflection in the Gibbs free energy of mixing curves. These points obey the following general conditions ... 

Let us again consider a solid mixture (A,B)N with a solvus field similar to the one outlined in the T-Xplot in figure 3. lOB, and let us analyze in detail the form of the Gibbs free energy of mixing curve in the zone between the two binodes (shaded area in figure 3.10A). 

Figure 3J2 Energy relationships between solvus and spinodal decompositions. (A) Portion of Gibbs free energy of mixing curve in zone between binodal (X ) and spinodal (X ) points. (B) Gibbs free energy variation as a consequence of compositional fluctuations around intermediate points X and X(2). ...

Figure 5.11 Gibbs free energy of mixing in binary join Mg2Si04-Ca2Si04 dXT = 600 °C and P = bar, calculated with a static interionic potential approach. Reprinted from G. Ottonello, Geochimica et Cosmochimica Acta, 3119-3135, copyright 1987, with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington 0X5 1GB, UK.

A more recent model (Ghiorso, 1984) is based on the binary interaction parameters of Thompson and Hovis (1979) for the NaAlSi30g-KAlSi30g join and on the experimental results of Newton et al. (1980), coupled with the A1 avoidance principle of Kerrick and Darken (1975) extended to the ternary field. Ghiorso (1984) expressed the excess Gibbs free energy of mixing in the form... 

Because two moles of O produce one mole of 0° and one of, the Gibbs free energy of mixing per mole of silicate melt is given by... 

Figure 6.2 Gibbs free energy of mixing in the quasi-chemical model of Toop and Samis (1962a,b), compared with values experimentally observed in Pb0-Si02 and Ca0-Si02 melts. Reprinted from Toop and Samis (1962b), with kind permission of ASM International, Materials Park, Ohio.

We have already seen that the degree of polymerization of the melt is controlled by the amount of silica in the system (see, for instance, figure 6.4). If we mix two fused salts with the same amount of silica and with cations of similar properties, the anion matrix is not modified by the mixing process and the Gibbs free energy of mixing arises entirely from mixing in the cation matrix—i.e.. 

If the heat capacity of a chemically complex melt can be obtained by a linear summation of the specific heat of the dissolved oxide constituents at all T (i.e., Stebbins-Carmichael model), the melt is by definition ideal. The addition of excess Gibbs free energy terms thus implies that the Stebbins-Carmichael model calculates only the ideal contribution to the Gibbs free energy of mixing. 

Calculation of the excess Gibbs free energy of mixing (third term on right side of eq. 6.78) involves only binary interactions. Although there is no multiple interaction model that can be reduced to the simple summation of binary interactions used here (cf Acree, 1984 see also section 3.10), this choice is more than adequate for the state of the art, which does not allow precise location of the miscibility gap in the chemical space of interest. 

Once the standard state potentials at the P and T of interest have been calculated (ix° = Gf for a pure single-component phase), the ideal and excess Gibbs free energy of mixing terms are easily obtained on the basis of the molar fractions of the various melt components and the binary interaction parameters listed in table 6.15 (cf eq. 6.78). 

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