Equilibrium, criteria

The chemical potential pi plays a vital role in both phase and chemical-reaction equilibria. However, the chemical potential exhibits certain unfortunate characteristics which discourage its use in the solution of practical problems. The Gibbs energy, and hence pi, is defined in relation to the internal energy and entropy, both primitive quantities for which absolute values are unknown. Moreover, pi approaches negative infinity when either P or Xi approaches zero. While these characteristics do not preclude the use of chemical potentials, the application of equilibrium criteria is facilitated by introduction of the fugacity, a quantity that takes the place of p. but which does not exhibit its less desirable characteristics. 

It should be kept in mind that an objective function which does not require any phase equilibrium calculations during each minimization step is the basis for a robust and efficient estimation method. The development of implicit objective functions is based on the phase equilibrium criteria (Englezos et al. 1990a). Finally, it should be noted that one important underlying assumption in applying ML estimation is that the model is capable of representing the data without any systematic deviation. Cubic equations of state compute equilibrium properties of fluid mixtures with a variable degree of success and hence the ML method should be used with caution. 

Only two of the four state variables measured in a binary VLE experiment are independent. Hence, one can arbitrarily select two as the independent variables and use the EoS and the phase equilibrium criteria to calculate values for the other two (dependent variables). Let Q, (i=l,2,...,N and j=l,2) be the independent variables. Then the dependent ones, g-, can be obtained from the phase equilibrium relationships (Modell and Reid, 1983) using the EoS. The relationship between the independent and dependent variables is nonlinear and is written as follows... 

For sub-critical isotherms (T < Tc), the parts of the isotherm where (dp/dV)T < 0 become unphysical, since this implies that the thermodynamic system has negative compressibility. At the particular reduced volumes where (dp/dV)T =0, (spinodal points that correspond to those discussed for solutions in the previous section. This breakdown of the van der Waals equation of state can be bypassed by allowing the system to become heterogeneous at equilibrium. The two phases formed at T[Pg.141]

Multicomponent diffusion, 1 43—46 Multicomponent mixtures, phase and chemical equilibrium criteria in, 24 675-678... 

Stoichiometric saturation defines equilibrium between an aqueous solution and homogeneous multi-component solid of fixed composition (10). At stoichiometric saturation the composition of the solid remains fixed even though the mineral is part of a continuous compositional series. Since, in this case, the composition of the solid is invariant, the solid may be treated as a one-component phase and Equation 6 is the only equilibrium criteria applicable. Equations 1 and 2 no longer apply at stoichiometric saturation because, owing to kinetic restrictions, the solid and saturated solution compositions are not free to change in establishing an equivalence of individual component chemical potentials between solid and aqueous solution. The equilibrium constant, K(x), is defined identically for both equilibrium and stoichiometric saturation. 

Nevertheless, rigorous applications of equilibrium criteria to rock-type and the minerals investigated can provide important information on mineral fractionations (Kohn and Valley 1998c Sharp 1995 Kitchen and Valley 1995). 

The weakness of this approach is that it deals with equilibrium criteria, whereas the situation in a furnace and certainly on a filament is highly dynamic. It must also assume some dissociation at all temperatures, and thus the appearance temperature becomes that at which the free metal is first detectable hence the parameter should be dependent upon the detection limit and concentration. Useful insights have been afforded by the application of thermodynamics, but clearly kinetic factors must also play a role. 

Three-component systems, or ternary systems, are fundamentally no different from two-component systems in terms of their thermodynamics. Phases in eqnilibrium must still meet the equilibrium criteria [Eqs. (2.14)-(2.16)], except that there may now be as many as five coexisting phases in eqnilibrinm with each other. The phase rule still... 

Thorstenson and Plummer (1977), in an elegant theoretical discussion (see section on The Fundamental Problems), discussed the equilibrium criteria applicable to a system composed of a two-component solid that is a member of a binary solid solution and an aqueous phase, depending on whether the solid reacts with fixed or variable composition. Because of kinetic restrictions, a solid may react with a fixed composition, even though it is a member of a continuous solid solution. Thorstenson and Plummer refer to equilibrium between such a solid and an aqueous phase as stoichiometric saturation. Because the solid reacts with fixed composition (reacts congruently), the chemical potentials of individual components cannot be equated between phases the solid reacts thermodynamically as a one-component phase. The variance of the system is reduced from two to one and, according to Thorstenson and Plummer, the only equilibrium constraint is IAP g. calcite = Keq(x>- where Keq(x) is the equilibrium constant for the solid, a function of... 

Garrels R.M. and Wollast R. (1978) Discussion of Equilibrium criteria for two-component solids reacting with fixed composition in an aqueous phase-example The magnesian calcites. Amer J. Sci. 278, 1469-1474. 

The mechanistic significance of Eq. 3.1, despite its great utility in mineral solubility studies, is, on the other hand, almost nil. No implication of reaction pathways, other than the postulate of stoichiometric control of aqueous-phase products, can be made on the basis of an overall reaction alone, since it features only enough species to satisfy minimal equilibrium criteria. Any number of additional kinetic species can intervene to govern the reaction pathways, which may be parallel, sequential, or a combination of these two, and the detailed interactions of the solid phase with the aqueous phase are often unlikely to be represented accurately by a spontaneous decomposition reaction like Eq. 3.1. As a case in point, the dissolution reaction of calcite (the forward reaction in Eq. 3.14) may be considered. Given the existence of protons and carbonate species in the aqueous-solution phase, at least three overall reactions more specific than Eq. 3.14 can be postulated to epitomize the detailed interactions of calcite with aqueous species 7,33,34... 

The equilibrium criteria for LLE are the same as for VLE, namely, uniformity of T, P, and of the fugacity /, for each chemical species tliroughout both phases. For LLE in a system of N species at uniform T and P, we denote the liquid phases by superscripts a and and write the equilibrium criteria as ... 

These are all functions of state of the system, just as U and 5, but give equilibrium criteria appropriate to different situations ... 

A where /,. and P are the fugacity of species i in the vapor mixture and the total pressure, respectively. From the thermodynamic equilibrium criteria, if the two phases of liquid and vapor are in a state of equilibrium, then ... 

Thus one would expect from a (6x6) matrix of the elastic stiffness coefficients (c,y) or compliance coefficients (sy) that there are 36 elastic constants. By the application of thermodynamic equilibrium criteria, cy (or Sjj) matrix can be shown to be symmetrical cy =cji and sy=Sji). Therefore there can be only 21 independent elastic constants for a completely anisotropic solid. These are known as first order elastic constants. For a crystalline material, periodicity brings in elements of symmetry. Therefore symmetry operation on a given crystal must be consistent with the representation of the elastic quantities. Thus for example in a cubic crystal the existence of 3C4 and 4C3 axes makes several of the elastic constants equal to each other or zero (zero when under symmetry operation cy becomes -cy,). As a result, cubic crystal has only three independent elastic constants (cu== C22=C33, C44= css= and Ci2=ci3= C2i=C23=C3i=C32). Cubic Symmetry is the highest that can be attained in a crystalline solid but a glass is even more symmetrical in the sense that it is completely isotropic. Therefore the independent elastic constants reduce further to only two, because C44=( c - C i)l2. 

The graphical method assumes an idealized absorption/stripping model that is defined in terms of essentially three components or groups of components a liquid, a gas, and a distributed component or solute. The liquid is assumed not to vaporize (i.e., it has a very low A -value), and the gas is assumed not to dissolve in the liquid (i.e., it has a very high A -value). The distributed component is the key component to be absorbed by the liquid or stripped by the gas. It distributes itself between the two phases to satisfy vapor-liquid equilibrium criteria (Chapter 1). 

---