Gibbs Free Energy of the Liquid Phase

Next we will derive the expression for the Gibbs free energy of the liquid phase, making the assumption that all the micelles are of the same size (that is, they are monodispersed). The monodispersity assumption, although supported by experimental data (Storm et ai, 1995) can be relaxed at the expense of computational complexity. 

As the sketch in Fig. 5.10 shows, the bulk liquid phase consists of micelles, monomeric asphalts and resins, and asphalt-free oil monomers in the bulk phase and in the shell. Let us denote asphaltenes and resins as the solute and the rest of the species (that is, asphalt-free oil species) as the solvent. Then the Gibbs free energy of the liquid phase, can be written as 

Later wc will represent the precipitated phase by L2. We assume a total of c components/pseudocomponents in the crude oil. The component indices for the asphaltenes and resins are c and (c — 1), respectively. Therefore, the number of species in the asphalt-free oil is c — 2). The 

The calculation of the Gibbs free energy of the solute components is more complex (see Pan and Firoozabadi, 1997a). We can divide into three parts, 

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

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FIGURE 7.26 For some substances and at certain pressures, the molar Gibbs free energy of the liquid phase might never lie lower than those of the other two phases. For such substances, the liquid is never the stable phase and, at constant pressure, the solid sublimes when the temperature is raised to the point of intersection of the solid and vapor lines. 

The Gibbs free energy of the liquid phase,, is obtained by adding Eqs. 

Solution Since amphiphiles have a polar head, they arc expected to coadsorb with resins onto the micellar-core surface. Therefore, one needs to modify both AG and the Gibbs free energy of the liquid phase L. In the following, we will briefly present the modifications to The standard Gibbs free energy of micellization, including the co-adsorption of the amphiphiles, can be expressed as,... 

To find how vapor pressure changes with temperature we make use of the fact that, when a liquid and its vapor are in equilibrium, there is no difference in the molar Gibbs free energies of the two phases ... 

Figure 8.30a shows how the molar Gibbs free energies of the liquid and solid phases of a pure solvent vary with temperature. The solid is the more stable phase... 

This equation can be written for the liquid and for the vapor. The ideal-gas term is the same in both phases because it depends only on temperature and pressure, which are the same in both phases. We conclude then that the residual Gibbs free energies of the two phases are also equal ... 

At high temperatures, Kokal et al. (1992), Hirschberg et al. (1984) and Godbole et al. (1995) observed the precipitation of a black-liquid mixture. A transition from solid to liquid state was observed by Storm et al. (1996) at above 330 K. Here we will formulate the Gibbs free energy of the precipitated phase as a liquid. The expression for the Gibbs free energy of the precipitated solid phase will be presented in Example 5.7. 

These expressions contain the differences between the Gibbs free energies of the liquid and the sohd phases in their pure standard states, i.e. these are the free energies of melting of the pure substances. They are, however, needed for temperatures other than T j, the equilibrium melting points of the pure substances. An approximate equation can be obtained in the following way. 

In practice, laborious modern calculations find the optimum distribution of elements between vapor and solid (or liquid) phases by minimizing the Gibbs free energy of the entire system. This calculation is accomplished in the following steps (Ebel, 2006) ... 

The melting point is defined as the temperature at which the liquid is in equilibrium with the solid so that the difference in Gibbs free energy between the two phases is zero. The entropy of fusion can then be expressed as. 

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