Equilibrium in a multiphase system

In this section we consider a system of a single substance in two or more uniform phases with distinctly different intensive properties. For instance, one phase might be a liquid and another a gas. We assume the phases are not separated by internal partitions, so that there is no constraint preventing the transfer of matter and energy among the phases. (A tall column of gas in a gravitational field is a different kind of system in which intensive properties of an equilibrium state vary continuously with elevation this case will be discussed in Sec. 8.1.4.) 

Phase a will be the reference phase. Since internal energy is extensive, we can write U = U + and dU = dI7 + dC/ . We assume any changes are slow 

This is an expression for the total differential of U when there are no constraints. 

We isolate the system by enclosing it in a rigid, stationary adiabatic container. The constraints needed to isolate the system, then, are given by the relations 

Each of these relations is an independent restriction that reduces the number of independent variables by one. When we substitute expressions for dU, dF , and d from these relations into Eq. 8.1.1, make the further substitution diS = dS — d and collect 

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It is presumed in this statement that equilibrium in a multiphase system implies uniformity of T and P throughout all phases. In certain situations, eg, osmotic equilibrium, pressure uniformity is not required, and equation 212 is then not a useful criterion. Here, however, it suffices. 

Equilibrium in a multiphase system implies thermal, mechanical, and material equilibrium. Thermal equilibrium requires uniformity of temperature throughout the system, and mechanical equilibrium requires uniformity of pressure. To find the criterion for material equilibrium, we treat a two-phase system and consider a transfer of dn moles from phase p to phase a. First, we regard each phase as a separate system. Because material enters or leaves these phases, they are open systems and we must use Eq. (4) to write their change in internal energy ... 

The chemical equilibrium in a multiphase system and the corresponding species concentrations in chemical separations are altered when chemical reactions take place. We will illustrate this now for gas-liquid equilibrium (as in gas absorption), vapor-liquid equilibrium (as in distillation), liquid-liquid equilibrium (as in solvent extraction), stationary phase-liquid equilibrium (as in ion exchange, chromatography and crystallization), surface adsorption equilibrium (as in foam fractionation) and Donnan equilibrium. 

The properties of a system at equilibrium do not depend on how the system arrived at equilibrium. Therefore, Eq. (5.1-5) is valid for any system at equilibrium, not only for a system that arrived at equilibrium under conditions of constant T and P. We call it the fundamental fact of phase equilibrium In a multiphase system at equilibrium the chemical potential of any substance has the same value in all phases in which it occurs. 

When considering heat transfer in a multiphase system, one can either treat each phase separately or the phases can be assumed to be in local equilibrium. In general, it is more complicated to treat each phase separately hence, we will use the local equilibrium approach that assumes all the phases have the same local average temperature. By performing an energy balance over a differential resin element one obtains,... 

For practical applications, such as for assessing the equilibrium distribution of a given compound in a multiphase system, it is most convenient to use a dimensionless air-solvent partition constant. This form uses molar concentrations in both phases. In this case, we denote the air-liquid partition constant as Kmf. Since Cia — pt /RT(Section 3.2), we then obtain ... 

DYNAMICS OF DISTRIBUTION The natural aqueous system is a complex multiphase system which contains dissolved chemicals as well as suspended solids. The metals present in such a system are likely to distribute themselves between the various components of the solid phase and the liquid phase. Such a distribution may attain (a) a true equilibrium or (b) follow a steady state condition. If an element in a system has attained a true equilibrium, the ratio of element concentrations in two phases (solid/liquid), in principle, must remain unchanged at any given temperature. The mathematical relation of metal concentrations in these two phases is governed by the Nernst distribution law (41) commonly called the partition coefficient (1 ) and is defined as = s) /a(l) where a(s) is the activity of metal ions associated with the solid phase and a( ) is the activity of metal ions associated with the liquid phase (dissolved). This behavior of element is a direct consequence of the dynamics of ionic distribution in a multiphase system. For dilute solution, which generally obeys Raoult s law (41) activity (a) of a metal ion can be substituted by its concentration, (c) moles L l or moles Kg i. This ratio (Kd) serves as a comparison for relative affinity of metal ions for various components-exchangeable, carbonate, oxide, organic-of the solid phase. Chemical potential which is a function of several variables controls the numerical values of Kd (41). 

The chapter divides in two in early sections we describe the behavior of nomeact-ing systems, while in later sections we deal with systems in which reactions occur. In 7.1 we combine the first and second laws to obtain criteria for identifying limitations on the directions of processes and for identifying equilibrium in closed multiphase systems. Then in 7.2 we develop the analogous relations for heat, work, and material transfers in open systems. With the material in 7.2 as a basis, we then present in 7.3 the thermodynamic criteria for equilibrium among phases. 

In a typical problem, multiple reactions are taking place in a multiphase system at fixed T and P, and we are to compute the equilibrium compositions of all phases. At this point, such calculations raise no new thermodynamic issues for example, for (R independent reactions occmrring among C species distributed between phases a and P, the problem is to solve the phase-equilibrium criteria... 

As depicted in Figure 2.2, the dissolved gaseous reactant has to overcome the resistance offered by the liquid-solid film before it reaches the solid catalyst surface where the reaction between the adsorbed reactive species takes place. The intrinsic capacity of a catalyst is realized when all mass transfer processes are at equilibrium. Therefore, it is required to know the rate of the solid-liquid mass transfer step. Such an estimate should reveal the relative importance of this step and also establish the controlling step in an overall process. In a multiphase system, the mass transfer between the liquid and particulate phases is considered to be good when there is an intimate mixing between the two phases. In the case of solid-liquid mass transfer, the minimum desirable condition is suspension of the solid in the liquid or in the gas-liquid dispersion as the case may be. 

In a multiphase formulation, such as an oil-in-water emulsion, preservative molecules will distribute themselves in an unstable equilibrium between the bulk aqueous phase and (i) the oil phase by partition, (ii) the surfactant micelles by solubilization, (iii) polymeric suspending agents and other solutes by competitive displacement of water of solvation, (iv) particulate and container surfaces by adsorption and, (v) any microorganisms present. Generally, the overall preservative efficiency can be related to the small proportion of preservative molecules remaining unbound in the bulk aqueous phase, although as this becomes depleted some slow re-equilibration between the components can be anticipated. The loss of neutral molecules into oil and micellar phases may be favoured over ionized species, although considerable variation in distribution is found between different systems. 

If the T and P of a multiphase system are constant, then the quantities capable of change are the individual mole numbers rf of the various chemical species i in the various phases p. In the absence of chemical reactions, which is assumed here, the ft may change only by interphase mass transfer, and not (because the system is closed) by the transfer of matter across the boundaries of the system. Hence, for phase equilibrium in a 7T-phase system, equation 212 is subject to a set of material balance constraints ... 

The equations derived for calculating the fractions of total i present in each phase at equilibrium in a two-phase system (Eqs. 3-62 and 3-63) can be easily extended to a multiphase system containing n phases (e.g., to a unit world ). If we pick one phase (denoted as phase 1) as the reference phase and if we use the partition constants of i between this phase and all other phases present in the system ... 

From the thermodynamic point of view, this is a multiphase system for which, at equilibrium, the Gibbs equation (A.20) must apply at each interface. Because there is no charge transfer in and out of layer (4) (an ideal insulator) the sandwich of the layers (3)/(4)/(5) also represents an ideal capacitor. It follows from the Gibbs equation that this system will reach electrostatic equilibrium when the switch Sw is closed. On the other hand, if the switch Sw remains open, another capacitor (l)/( )/(6) is formed, thus violating the one-capacitor rule. The signifies the undefined nature of such a capacitor. The open switch situation is equivalent to operation without a reference electrode (or a signal return). Acceptable equilibrium electrostatic conditions would be reached only if the second capacitor had a defined and invariable geometry. 

Thus, looking at the equilibrium phase diagram and knowing the physical-chemical properties of the elemets A and B and their compounds, it is possible to draw certain conclusions concerning the sequence of compound-layer formation in a multiphase binary system. It must be remembered, however, that any predictions based on the above-mentioned or other criteria hitherto proposed are only weak correlations, rather than the precise rules. As both the researcher and technologist are always interested in knowing the sequence of occurrence of chemical compounds in a particular reaction couple, they can hardly be satisfied even with a correlation valid in 99 out of 100 cases, because it remains unknown whether this couple falls in the range of those 99 or is the only exception. Further theoretical work in this direction is badly needed. 

Ordinarily, the system may consist of several phases, whose interior in the state of equilibrium is homogeneous throughout its extent. The system, if composed for instance of only liquid water, is a single phase and if made up for instance of liquid water and water vapor, it is a two phase system. The single phase system is frequently called a homogeneous system, and a multiphase system is called heterogeneous. 

The pseudo-Henry s law coefficient has no influence on the rate constant of the reaction because it is not a rate-but an equilibrium-constant. The rate constant in the multiphase system is only governed by the liquid water content and the concentrations of the oxidant and the H+ Ions. 

The Gibbs phase rule gives the degrees of freedom of a multiphase system in equilibrium, or the number of intensive (size-independent) system variables that must be specified before the others can be determined. 

The unit operation of erystallization is governed by some very complex interacting variables. It is a simultaneous heat and mass transfer process with a strong dependence on fluid and particle mechanics. It takes place in a multiphase, multicomponent system. It is concerned with particulate solids whose size and size distribution, both incapable of unique definition, vary with time. The solids are suspended in a solution which can fluctuate between a so-called metastable equilibrium and a labile state, and the solution composition can also vary with time. The nucleation and growth kinetics, the governing processes in this operation, can often be profoundly influenced by mere traces of impurity in the system a few parts per million may alter the crystalline product beyond all recognition. 

Pure substances may occur in a variety of phases, depending on the boundary conditions. Thus, increasing the temperature of a solid at constant pressure causes its fusion and finally its vaporization to a gas. In pure substances, phases correspond to the states of aggregation. In a heterogeneous system, on the other hand, a number of phases in the same state of a regation may coexist (see earlier examples). Here, we shall describe the conditions under which a multiphase system is in equilibrium in other words, when does a homogeneous system dissociate into different phases. ... 

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