Gibbs free energy convexity

The tools for calculating the equilibrium point of a chemical reaction arise from the definition of the chemical potential. If temperature and pressure are fixed, the equilibrium point of a reaction is the point at which the Gibbs free energy function G is at its minimum (Fig. 3.1). As with any convex-upward function, finding the minimum G is a matter of determining the point at which its derivative vanishes. 

Linear regions (constant slope, hence constant potentials) define the composition interval over which a two-phase assemblage is stable. Because the minimum Gibbs free energy curve of the system is never convex, the chemical potential of any component will always increase with the increase of its molar proportion in the system. 

Note it always has this exact shape—that of an upward-opening convex curve that passes through zero at 0 and 1. If the mixture were ideal, this curve plus that averaging the pure species Gibbs free energies would assume this upward-opening convex shape. 

We used infinite-dilution activity coefficients of 10 and 20 to create Fig. 6. Both are greater than 9, so we should expect the Margules equations to predict liquid/liquid behavior. Water and toluene have infinite-dilution activity coefficients in the thousands. They really dislike each other and break into relatively pure phases. If we examine the total Gibbs free energy curve, we gain the impression that the curve is totally convex-upward however, there is a slight downward move at the extremes because of the infinite downward slope of the mixing term at the extreme compositions. The two liquid phases are almost, but not quite pure. 

M. Feinberg. On the convexity of the ideal gas Gibbs free energy function. Personal communication, 1998,... 

A note on the Wilson equation This equation results in a convex graph for the Gibbs free energy of mixing for all values of the parameters A, and A, therefore, it is not capable of predicting phase splitting. All other models discussed in Chapter 12 can produce concave shapes and thus may be used to describe partially miscible systems. 

The first and second conditions ensure the existence of the thermodynamic Gibbs free energy function or, using the mathematical term, the convex Lyapunov function for kinetic equations. The Lyapunov function is a strictly positive function with a nonpositive derivative. The one exception to this definition is that at the equilibrium point, the Lyapunov function equals zero. In physicochemical sciences, the Gibbs free energy is an extremely important Lyapunov function for understanding the stability of equilibria. 

Free energy bounds can be established via the Gibbs-Bogoliubov inequality [72], which follows from Eq. (2.6) by considering the convexity of the exponential function... 

Under conditions of very slow growth, the ultimate crystal shape is determined strictly by thermodynamics. Under such conditions, the faces appearing on the crystal correspond to the smallest convex polyhedron having minimum surface free energy. Gibbs... 

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