The Gibbs Phase Rule for Multicomponent Systems

We encountered the Gibbs phase rule and phase diagrams in Chap. 8 in connection with single-substance systems. The present chapter derives the full version of the Gibbs phase rule for multicomponent systems. It then discusses phase diagrams for some representative t) es of multicomponent systems, and shows how they are related to the phase rule and to equilibrium concepts developed in Chaps. 11 and 12. 

In Sec. 8.1.7, the Gibbs phase rule for a pure substance was written F = 3 — P. We now consider a system of more than one substance and more than one phase in an equilibrium state. The phase rule assumes the system is at thermal and mechanical equihbrium. We shall assume furthermore that in addition to the temperature and pressure, the only other state functions needed to describe the state are the amounts of the species in each phase this means for instance that surface effects are ignored. 

The derivations to follow will show that phase rule may be written either in the form 

If we subdivide a phase, that does not change the number of phases P. That is, we treat noncontiguous regions of the system that have identical intensive properties as parts of the same phase. 

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CHAPTER 13 THE PHASE RULE AND PHASE DIAGRAMS 13.1 The Gibbs Phase rule for Multicomponent Systems... 

An understanding of the Gibbs phase rule for multicomponent systems allows us to consider specific multicomponent systems. We will focus on two-component systems for illustration, although the concepts are applicable to systems with more than two components. 

The application of the Gibbs phase rule to multicomponent systems containing c components may he done by extending the treatment used for the unary system in Section 1.2.1.1. Again let ip represent the number of... 

The Gibbs phase rule for a multicomponent system to be described in Sec. 13.1 shows that a two-component, two-phase system at equilibrium has only two independent intensive variables. Thus at a given temperature and pressure, the mole fraction compositions of both phases are fixed the compositions depend only on the identity of the substances and the temperature and pressure. 

The lacking special description of the Gibbs phase rule in MEIS that should be met automatically in case of its validity is very important for solution of many problems on the analysis of multiphase, multicomponent systems. Indeed, without information (at least complete enough) on the process mechanism (for coal combustion, for example, it may consist of thousands of stages), it is impossible to specify the number of independent reactions and the number of phases. Prior to calculations it is difficult to evaluate, concentrations of what substances will turn out to be negligibly low, i.e., the dimensionality of the studied system. Besides, note that the MEIS application leads to departure from the Gibbs classical definition of the notion of a system component and its interpretation not as an individual substance, but only as part of this substance that is contained in any one phase. For example, if water in the reactive mixture is in gas and liquid phases, its corresponding phase contents represent different parameters of the considered system. Such an expansion of the space of variables in the problem solved facilitates its reduction to the CP problems. 

Distinguish between intensive and extensive variables, giving examples of each. Use the Gibbs phase rule to determine the number of degrees of freedom for a multicomponent multiphase system at equilibrium, and state the meaning of the value you calculate in terms of the system s intensive variables. Specify a feasible set of intensive variables that will enable the remaining intensive variables to be calculated. 

When multicomponent gas and liquid phases are in equilibrium, a limited number of intensive system variables may be specified arbitrarily (the number is given by the Gibbs phase rule), and the remaining variables can then be determined using equilibrium relationships for the distribution of components between the two phases. In this section we define several such relationships and illustrate how they are used in the solution of material balance problems. 

If you apply the Gibbs phase rule to a multicomponent gas-liquid system at equilibrium, you will discover that the compositions of the two phases at a given temperature and pressure are not independent. Once the composition of one of the phases is specified (in terms of mole fractions. mass fractions, concentrations, or. for the vapor phase, partial pressures), the composition of the other phase is fixed and, in principle, can be determined from physical properties of the system components. 

In 1926 Kohnstamm extended the Gibbs phase rule to encompass the appearance of critical points in one- and multicomponent systems. He assumed that the critical point may be considered as a specific, additional phase. If / phases coexist, and next become critical, p l meniscuses disappear in a solution consisting of c components. Hence, the Gibbs phase mle supplemented by the critical phase has the following form =c- p + p-l) + 2 = c-2p + > Consequently, for / -critical point at least c = 2p- component system is required. This yields for = 0, i.e. a single critical point, following conditions ... 

Although occasionally papers appear speaking of the inapplicability of Gibbs phase rule [Li, 1994, 13] or beyond the Gibbs phase rule [Mladek et al, 2007], this invariably means no more than that one of the ceteris paribus conditions Gibbs already mentioned is not fulfilled for example, the phase rule doesn t cover systems in which rigid semi-permeable walls allow the development of pressure differences in the system. Gibbs explicitly allows for the possible presence of other thermodynamic fields. An extended phase rule has been proposed for, inter alia, capillary systems (in which the number and curvature of interfaces/phases play a role), multicomponent multiphase systems for which relative phase sizes are relevant [Van Poolen, 1990], colloid systems (for which, even if in equilibrium, it is not always easy to say how many phases are present), unusual crystalline materials, and more. 

A straightforward, but tedious, route to obtain information of vapor-liquid and liquid-liquid coexistence lines for polymeric fluids is to perform multiple simulations in either the canonical or the isobaric-isothermal ensemble and to measure the chemical potential of all species. The simulation volumes or external pressures (and for multicomponent systems also the compositions) are then systematically changed to find the conditions that satisfy Gibbs phase coexistence rule. Since calculations of the chemical potentials are required, these techniques are often referred to as NVT- or NPT- methods. For the special case of polymeric fluids, these methods can be used very advantageously in combination with the incremental potential algorithm. Thus, phase equilibria can be obtained under conditions and for chain lengths where chemical potentials cannot be reliably obtained with unbiased or biased insertion methods, but can still be estimated using the incremental chemical potential ansatz [47-50]. 

If a multiphase multicomponent system is to be at equilibrium (no change with time of the intensive variables) obviously temperature and pressure must be the same for all phases and also the chemical compositions (mole fractions of each constituent). In any given phase there are (C—1) independent mole fractions (their sum is unity by definition), so there are P.(C—1) composition variables involved and thus [P.(C—1) -1-2] intensive variables in total. But if chemical equUibrium in all phases simultaneously is to hold, the chemical potential of each constituent (a function of the composition) must be the same in each phase thus there are C.(P—1) independent constraints on the composition variables arising from the equilibrium condition (the chemical potential in one of the phases is used as the reference standard for the other phases). Thus F = [P.(C—1) -1-2] — [C.(P—1)] =C—P-F2. This is the famous Gibbs Phase Rule. 

The mathematical basis of classic thermodynamics was developed by J. Willard Gibbs in his essay [1], On the Equilibrium of Heterogeneous Substances, which builds on the earlier work of Kelvin, Clausius, and Helmholtz, among others. In particular, he derived the phase mle, which describes the conditions of equilibrium for multiphase, multicomponent systems, which are so important to the geologist and to the materials scientist. In this chapter, we will present a derivation of the phase rule and apply the result to several examples. 

We show here the derivation of the phase rule as done by Gibbs in his original paper [3, pp. 55-353]. In a multicomponent system for a homogeneous phase the... 

In the general case of size-composition-dependent surface energy contribution for equilibrium two-phase state, one must solve the above given system of equations with complementary parameters and terms da/dr and da/dC. Furthermore, the rule may be applied to a multicomponent system as well when a new phase is not determined by strong stoichiometric composition that is, there exists the solubility interval on the diagram Gibbs free energy density-concentration (Ag(Q — C). As was mentioned before in the presented case. Equation 13.A.4 is applied to nucleation and separation of nanoparticles in which the composition of the new phase is a function of size. 

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