Gibbs Free Energy Surface Analysis

Now consider the mixing of components 1 and 2 at constant temperature and pressure. The molar Gibbs free energy of mixing, can 

In Example 4.11, we will illustrate how to calculate over the whole range of composition (that is, 0 2 1), making the assumption that over the whole range, the mixture stays in the hypothetical homogeneous single-phase state. 

The above relationship can be derived by taking the derivative of Eq. (4.39) with respect to X2 and using the Gibbs-Duhem equation ZLi = 0 at constant temperature and pressure. Finally, by 

The proof of the equality of chemical potentials is simple. From Fig. 4.5a, 

Gibbs free energy of the two-phase mixture is A 2- From 23 = + 

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Equilibrium, stability, and criticality arc important concepts that are closely related. In this chapter, after the formulation of simple methods for phase-equilibria calculations, the concepts of stability and criticality are introduced, and the application of the Gibbs free energy surface analysis to phase-equilibrium calculations is demonstrated. Then the stability and criticality concepts are presented in detail. These concepts are useful for a broad range of problems in engineering and physics. 

Gibbs free energies of water sorption, AG "(/l), can be extracted from isopiestic vapor sorption isotherms this analysis shows that AG (T) < AG", where AG" = -44.7 kj moH is the Gibbs free energy for vapor sorption at a free water surface at ambient condihons. Water absorbed by the membrane is therefore more strongly bound than water at a free bulk water surface this affirms the hydrophilic nature of water sorption in PEMs. 

The aforementioned analysis indicates that the entropic contribution to the Gibbs free energy of adsorption inereases with inereasing temperature. These results are in harmony with the notion that protein adsorption at hydrophobic surfaces is entropieally driven [13,14], and such entropically driven, hydrophobic interactions between the protein and the silicon surface are favored with increasing temperature. 

Thermal analysis data on lamellar crystals of polyethylene over a wide range of thicknesses are plotted in Fig. 2.90. The Gibbs-Thomson equation is a good mathematical description of the observed straight line and can be used to calculate the equilibrium melting temperature by setting C = (t ° = 414.2 K). Also, the ratio of the surface free energy to the heat of fusion can be obtained from the equation. 

It is now a simple matter to derive Eq. (5), at bottom of Fig. 4.28, from the equality of the two melting terms of Eq. (4). The measurable lowering in melting temperature from equilibrium, AT, is related to the surface free energy a and the lamellar thickness t. If one of the two latter two quantities is known, the other can be calculated from thermal analysis. Equation (5) is also called the Gibbs-Thomson equation. 

In the section below, we discuss the correlation between the critical nucleus and surface energy (23). A more complete analysis of precipitation must take into account the chemical potential of the nucleus formed in equilibrium with the reactional media. In such a condition, the nucleus possesses free Gibbs energy, as follows [23,98,108,124] ... 

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