Mixing Gibbs free energy

The mixing Gibbs free energy is given by (see Example 1.4, Chapter 1)... 

A mix Gibbs free energy of mixing for a binary polymer system... 

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.

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... 

A solution is a single-phase mixture of more than one compound, and the driving force for its spontaneous formation from the pure compounds at constant T and p is the negative Gibbs free energy change of the mixing process, —AG, as... 

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... 

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... 

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... 

The structure of hydrogels that do not contain ionic moieties can be analyzed by the Flory Rehner theory (Flory and Rehner 1943a). This combination of thermodynamic and elasticity theories states that a cross-linked polymer gel which is immersed in a fluid and allowed to reach equilibrium with its surroundings is subject only to two opposing forces, the thermodynamic force of mixing and the retractive force of the polymer chains. At equilibrium, these two forces are equal. Equation (1) describes the physical situation in terms of the Gibbs free energy. 

In principle, Gibbs free energies of transfer for trihalides can be obtained from solubilities in water and in nonaqueous or mixed aqueous solutions. However, there are two major obstacles here. The first is the prevalence of hydrates and solvates. This may complicate the calculation of AGtr(LnX3) values, for application of the standard formula connecting AGt, with solubilities requires that the composition of the solid phase be the same in equilibrium with the two solvent media in question. The other major hurdle is that solubilities of the trichlorides, tribromides, and triiodides in water are so high that knowledge of activity coefficients, which indeed are known to be far from unity 4b), is essential (201). These can, indeed, be measured, but such measurements require much time, care, and patience. 

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. 

Equations 3.121 and 3.122 distinguish the bulk Gibbs free energy of the mixture ( mixture) from the Gibbs free energy term involved in the mixing procedure... 

If mixture (A,B)N is ideal, mixing will take place without any heat loss or heat production. Moreover, the two cations will be fully interchangeable in other words, if they occur in the same amounts in the mixture, we will have an equal opportunity of finding A or B over the same structural position. The Gibbs free energy term involved in the mixing process is... 

The Gibbs free energy of mixing can also be expressed as... 

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. 

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... 

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