Gibbs free energy, excess

Real solutions are rarely completely athermal, even when there is considerable similarity between the nature of the molecules. For cases in which some energy effects must be taken into account, Flory introduced an additional term into the expression for excess Gibbs free energy. Adapting the format of the Scatchard-Hildebrand equation, the additional contribution to the excess Gibbs free energy is assumed to be of the form ... 

J. B. Ott, K. N. Marsh and R. H. Stokes. "Excess Enthalpies. Excess Gibbs Free Energies, and Excess Volumes for (Cyclohexane + //-Hexane). and Excess Gibbs Free Energies and Excess Volumes for (Cyclohexane + Methylcyclohexane) at 298.15 and 308.15 K". J.Chem. Thermodyn., 12, 1139-1148 (1980). 

For a detailed discussion of the calculation of activities (and excess Gibbs free energies) from freezing point measurements, see R. L. Snow. J. B. Ott. J. R. Goates. K. N. Marsh, S. O Shea, and R. N. Stokes. "(Solid + Liquid) and (Vapor + Liquid) Phase Equilibria and Excess Enthalpies for (Benzene + //-Tetradecane), (Benzene + //-Hexadecane). (Cyclohexane + //-Tetradecane), and (Cyclohexane +//-Hexadecane) at 293.15, 298.15, and... 

Since these mixing processes occur at constant pressure, // is the heat evolved or absorbed upon mixing. It is usually measured in a mixing calorimeter. The excess Gibbs free energy, is usually obtained from phase equilibria measurements that yield the activity of each component in the mixtureb and S is then obtained from equation (7.17). The excess volumes are usually obtained... 

Liu, A. Beck, T. L., Determination of excess Gibbs free energy of quantum mixtures by MC path integral simulations, Mol. Phys. 1995, 86, 225-233... 

For correlation of solubility, the correct thermodynamic quantities for correlation are the activity coefficient y, or the excess Gibbs free energy AG, as discussed by Pierotti et al. (1959) and Tsonopoulos and Prausnitz (1971). Examples of such correlations are given below. 

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 ... 

For the industrially important class of mixed solvent, electrolyte systems, the Pitzer equation is not useful because its parameters are unknown functions of solvent composition. A local composition model is developed for these systems which assumes that the excess Gibbs free energy is the sum of two contributions, one resulting from long-range forces between ions and the other from short-range forces between all species. 

Garriga, R., Sanchez, F., Perez, P., and Gracia, M. Excess Gibbs free energies at eight temperatnres and excess enthalpies and volnmes at 7 = 198.15 for bntanenitrile + 2-bntanol, J. Chem. Eng. Data, 42(l) 78-83, 1997. 

Carpenter, 1988). Because the excess Gibbs free energy of transition must always be at a minimum with respect to the macroscopic order parameter Q, i.e. ... 

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

The excess Gibbs free energy of mixing is given by... 

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. 

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

The Kohler model is a general model based on linear combination of the binary interactions among the components in a mixture, calculated as if they were present in binary combination (relative proportions) and then normalized to the actual molar concentrations in the multicomponent system. The generalized expression for the excess Gibbs free energy is... 

Table 5.41 Macroscopic interaction parameters for C2 c quadrilateral pyroxenes. Resulting excess Gibbs free energies are in J/mole (from Ottonello, 1992).

Salje (1985) interpreted overlapping (displacive plus Al-Si substitutional) phase transitions in albite in the light of Landau theory (see section 2.8.1), assigning two distinct order parameters Q n and to displacive and substitutional disorder and expanding the excess Gibbs free energy of transition in the appropriate Landau form ... 

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... 

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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