Minimal free energy, equilibrium phase

If the phase separation is possible, the free energy of phase separation is negative and characterized by a deep minimum. Because of this the unknown variables of the equation, which are necessary for constructing phase diagrams, may be found by minimization of the fimction Agpg. This term, as distinct from the free energy of mixing, allows in the closed form the condition of the phase equilibrium (bimodal equation) to be recorded for... 

R.C. Oliver et al, USDeptCom, Office Tech-Serv ..AD 265822,(1961) CA 60, 10466 (1969) Metal additives for solid proplnts formulas for calculating specific impulse and other proplnt performance parameters are given. A mathematical treatment of the free-energy minimization procedure for equilibrium compn calcns is provided. The treatment is extended to include ionized species and mixing of condensed phases. Sources and techniques for thermodynamic-property calcns are also discussed... 

Smit et al. [19] used the partition function given by (10.4) and a free energy minimization procedure to show that, for a system with a first-order phase transition, the two regions in a Gibbs ensemble simulation are expected to reach the correct equilibrium densities. 

The most important property of a liquid-gas interface is its surface energy. Surface tension arises at the boundary because of the grossly unequal attractive forces of the liquid subphase for molecules at its surface relative to their attraction by the molecules of the gas phase. These forces tend to pull the surface molecules into the interior of the liquid phase and, as a consequence, cause liquids to minimize their surface area. If equilibrium thermodynamics apply, the surface tension 7 is the partial derivative of the Helmholtz free energy of the system with respect to the area of the interface—when all other conditions are held constant. For a phase surface, the corresponding relation of 7 to Gibbs free energy G and surface area A is shown in eq. [ 1 ]. 

Stripping of chlorine from hydroxides such as Cl2Sn(OH)2 could eventually lead to gas-phase SnO or Sn02. However, at the relatively low temperatures typical of tin oxide CVD ( 873-973 K), we do not expect these oxides to form, based on the equilibrium calculations described above. Thus, the formation of tin hydroxides is not only thermodynamically favored (i.e., based on minimization of the Gibbs free energy), but there are also exothermic reaction pathways that we expect to be kinetically favorable. The primary tin carrier in the CVD process could therefore be a tin hydroxide. Complete conversion to Sn02 would most likely occur via reactions on the surface. 

Rahaman and Hatton [152] developed a thermodynamic model for the prediction of the sizes of the protein filled and unfilled RMs as a function of system parameters such as ionic strength, protein charge, and size, Wq and protein concentration for both phase transfer and injection techniques. The important assumptions considered include (i) reverse micellar population is bidisperse, (ii) charge distribution is uniform, (iii) electrostatic interactions within a micelle and between a protein and micellar interface are represented by nonlinear Poisson-Boltzmann equation, (iv) the equilibrium micellar radii are assumed to be those that minimize the system free energy, and (v) water transferred between the two phases is too small to change chemical potential. 

Chemical Equilibrium The chemical equilibrium approach is more complex computationally than applying the assumption of an infinitely fast reaction. The equilibrium composition of a multicomponent system is estimated by minimizing the Gibbs free energy of the system. For a gas-phase system with K chemical species, the total Gibbs free energy may be written as... 

In a recent paper Shapiro Shapley [4] have considered the problem of the uniqueness of equilibrium of systems of reactions in several phases in great detail. The computation of equilibrium compositions by direct minimization of the Gibbs free energy function has proved a valuable tool in the discussion of very complex systems and it is important to show that this minimum is unique and achieved under the same conditions that satisfy the mass action laws. This is what Shapiro Shapley have done Sellers has suggested some improvements [5]. 

We can now distinguish three stages in the development of thermodynamics. First, there is the equilibrium stage in which the forces and the consequent flows vanish. It is under those conditions that we have equilibrium phase transitions such as solid to liquid and liquid to vapor. The structures that arise in such phenomenon, as for instance in a crystal, can be understood in terms of the minimization of the well-known free energy F. We have... 

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