Equilibrium Relations Between Phases

2 EQUILIBRIUM RELATIONS BETWEEN PHASES 10.2A Phase Rule and Equilibrium 

In order to predict the concentration of a solute in each of two phases in equilibrium, experimental equilibrium data must be available. Also, if the two phases are not at equilibrium, the rate of mass transfer is proportional to the driving force, which is the departure from equilibrium. In all cases involving equilibria, two phases are involved, such as gas-liquid or liquid-liquid. The important variables affecting the equilibrium of a solute are temperature, pressure, and concentration. 

The equilibrium between two phases in a given situation is restricted by the phase 

This means that there are 3 degrees of freedom. If the total pressure and the temperature are set, only one variable is left that can be arbitrarily set. If the mole fraction composition of COj [A) in the liquid phase is set, the mole fraction composition or pressure in the gas phase is automatically determined. 

The phase rule does not tell us the partial pressure in equilibrium with the selected x,. The value of must be determined experimentally. The two phases can, of course, be gas-liquid, liquid-solid, and so on. For example, the equilibrium distribution of acetic acid between a water phase and an isopropyl ether phase has been determined experimentally for various conditions. 

---

Equilibrium relations between phases P and S, and e and are also needed. These relations can be obtained using the electrochemical reactions given in Eqs (3) and (4). 

In this approach, it is assumed that turbulence dies out at the interface and that a laminar layer exists in each of the two fluids. Outside the laminar layer, turbulent eddies supplement the action caused by the random movement of the molecules, and the resistance to transfer becomes progressively smaller. For equimolecular counterdiffusion the concentration gradient is therefore linear close to the interface, and gradually becomes less at greater distances as shown in Figure 10.5 by the full lines ABC and DEF. The basis of the theory is the assumption that the zones in which the resistance to transfer lies can be replaced by two hypothetical layers, one on each side of the interface, in which the transfer is entirely by molecular diffusion. The concentration gradient is therefore linear in each of these layers and zero outside. The broken lines AGC and DHF indicate the hypothetical concentration distributions, and the thicknesses of the two films arc L and L2. Equilibrium is assumed to exist at the interface and therefore the relative positions of the points C and D are determined by the equilibrium relation between the phases. In Figure 10.5, the scales are not necessarily the same on the two sides of the interface. 

The problems relating to mass transfer may be elucidated out by two clear-cut yet different methods one using the concept of equilibrium stages, and the other built on diffusional rate processes. The selection of a method depends on the type of device in which the operation is performed. Distillation (and sometimes also liquid extraction) are carried out in equipment such as mixer settler trains, diffusion batteries, or plate towers which contain a series of discrete processing units, and problems in these spheres are usually solved by equilibrium-stage calculation. Gas absorption and other operations which are performed in packed towers and similar devices are usually dealt with utilizing the concept of a diffusional process. All mass transfer calculations, however, involve a knowledge of the equilibrium relationships between phases. 

In any event, there is always a strict, equilibrium relation between the charge density, qM, on the electrode sur ce and the total potential difference, E, between the bulk phases of electrode and solution. This relation is often characterized by the differential double-layer capacity, Cd, defined as... 

Figure 6.1 shows equilibrium relations between stable (A) and metastable (B) carbonate minerals. The boundaries between stability fields are most easily obtained by consideration of the mass action equations representing mineral compatibilities in terms of the variables shown in Figure 6.1. For example, consider the phase boundaries in Figure 6.1 A the reactions and the equilibrium constants for these boundaries in terms of the ratio of activity of Ca2+ to activity of Mg2+ and PCO2 are 3s follows ...

Sorption of VOCs involves the processes of adsorption and partitioning. Partitioning is the incorporation of the VOC into the natural organic matter associated with the solid and is analogous to the dissolution of an organic compound into an organic solvent. Adsorption is the formation of a chemical or physical bond between the VOC and the mineral surface of a solid particle (Rathbun, 1998). The equilibrium relation between aqueous and solid phase concentrations then is expressed as... 

Here, assume that in the range of compositions involved, the thermodynamically phase equilibrium relations between rich and lean streams are linear, and concern with the operating temperature T and pressure P, then we can obtain phase equilibrium equations as Eq. (1). 

Physical chemists distinguish between adsorption and absorption. Adsorption is a surface phenomenon. Consider a solid or liquid phase (the adsorbent), in contact with another, fluid, phase. Molecules present in the fluid phase may now adsorb onto the interface between the phases, i.e., form a (usually monomolecular) layer of adsorbate. This is discussed in more detail in Section 10.2. The amount adsorbed is governed by the activity of the adsorbate. For any combination of adsorbate, adsorbent, and temperature, an adsorption isotherm can be determined, i.e., a curve that gives the equilibrium relation between the amount adsorbed per unit surface area, and the activity of the adsorbate. Powdered solid materials in contact... 

Whereas the temperature-dependence of the free energy of a phase is related to its entropy, the pressure dependence is related to its volume. As shown in Section 4.3, (dG/dP)r = V, and the variation in the equilibrium position between phases (as represented by the lines in Fig. 4.3 depends on the volume change associated with the phase transition. Thus as AV for melting is small compared with the volume change associated with vaporization, the melting point is very much less sensitive to pressure than the boiling point. 

By assuming an equilibrium adsorption desorption of component A between the internal surface and the intraparticle gas, equations 7.15 and 7.18 are combined to give the following equilibrium relation between the surface concentration and the gas phase concentration ... 

Fig. 17. PPO4 versus pS for upper 8 cm at FOAM, NWC, and DEEP where pS was detectable. The line drawn is the equilibrium relation between vivianite (Fe3(P04)2 8H2O) and mackinawite (FeS). , FOAM O. NWC A, DEEP. All seasons plotted. The formation of sulfides is generally favored, but regions exist near the sediment-water interface where coexistence of phases or phosphate dominance are possible.

The relation between phase separation and viscosity has been studied as a function of time during which the scale of phase separation increases. Under these circumstances, the viscosity has been observed to increase by as much as five orders of magnitude during an isothermal heat treatment for times of several hundred hours. The viscosity initially changes rapidly as the connectivity of the structure and the compositions of the equilibrium phases approach their final values, and then more slowly as coarsening, or growth in the scale of the microstructure, occurs. 

Henry s law. Often the equilibrium relation between in the gas phase andx,< be expressed by a straight-line Henry s law equation at low concentrations. 

In membrane processes with two gas phases and a solid membrane, similar equations can be written for the case shown in Fig. 13.2-lb. The equilibrium relation between the solid and gas phases is given by... 

The most important conception in adsorption science is that named as the adsorption isotherm . It is the equilibrium relation between the quantity of the adsorbed material and the pressure or concentration in the bulk fluid phase at constant temperature. 

Define the rate of the different processes in terms of the state variables and rate parameters, and introduce the necessary equations of state and equilibrium relations between the different phases. Note Usually, equilibrium relations between certain variables are used instead of rates as an approximation when the rate is quite high and the process reaches equilibrium quickly. 

The overall molar balance we used will not be adequate because the equilibrium relation between the two phases cannot be used. In the present case of nonequilibrium stages, we must introduce the mass transfer rate between the two phases, as shown in Figure 6.18. In the figure, the term RMT is the rate of mass transfer of the component A from the vapor phase. It can be expressed as,... 

For a phenomenological description of chromatographic separation, the concept of theoretical plate is often used. Chromatographic column is considered to consist of a large number of theoretical plates, in each of which equilibrium relations between fluid and particle phases hold (Fig. 10.1). Height equivalent to theoretical plate (HETP) and number of theoretical plate (NTP) are then related to column length z as... 

In the majority of the kinetic studies, the reaction of sulfur dioxide with calcium oxide was considered as first order with respect to SO2 concentration. On the other hand, in a recent study of Borgwardt and Bruce (1986) an equilibrium relation between the partial pressure of sulfur dioxide in the gas phase and concentration of sulfur dioxide on the product layer surface was proposed. 

Although the relations given have been developed for any gas-liquid system, they are also valid for any vapor-liquid system. The only difference is that Raoult s law is to be used instead of Henry s law as the equilibrium relation between the phases. For example, consider transfer of species i from vapor phase to liquid phase, Xigb to Xub, with the overall mass-transfer coefficient being expressed in terms of the vapor phase, i.e. Kxg. 

In order to use any of the aforementioned operations, the equilibrium relations between liquids and vapors must be ascertained. In other words, the distribution of a component in the two phases after no further interchange takes place must be known. Although Dalton s law of partial pressures, Henry s law of gas solubility, and Raoult s law of vapor-liquid equilibrium are not exact, they are used extensively because complete data are not available. 

If the equilibrium relation y° = F Xi) is sufficiently simple, e.g., if a plot of yfversus Xi is a straight hne, not necessarily through the origin, the rate of transfer is proportional to the difference between the bulk concentration in one phase and the concentration (in that same phase) which would be in equilibrium with the bulk concentration in the second phase. One such difference isy — y°, and another is x° — x. In this case, there is no need to solve for the interfacial compositions, as may be seen from the following derivation. 

The efficiencies which may be obtained can consequently be calculated by simple stoichiometry from the equilibrium data. In the ease of countercurrent-packed columns, the solute can theoretically be completely extracted, but equilibrium is not always reached because of the poorer contact between the phases. The rate of solute transfer between phases governs the operation, and the analytical treatment of the performance of such equipment follows closely the methods employed for gas absorption. In the ease of two immiscible liquids, the equilibrium concentrations of a third component in each of the two phases are ordinarily related as follows ... 

The relation between the potential-composition data for these two systems under equilibrium conditions is shown in Fig. 13. It is seen that the phase Li2hSn (Lil3Sn5) is stable over a potential range that includes the upper two-phase reconstitution reaction plateau in the lithium-silicon system. Therefore, lithium can react with Si to form the phase Li, 7 S i... 

In the case of coupled heterogeneous catalytic reactions the form of the concentration curves of analytically determined gaseous or liquid components in the course of the reaction strongly depends on the relation between the rates of adsorption-desorption steps and the rates of surface chemical reactions. This is associated with the fact that even in the case of the simplest consecutive or parallel catalytic reaction the elementary steps (adsorption, surface reaction, and desorption) always constitute a system of both consecutive and parallel processes. If the slowest, i.e. ratedetermining steps, are surface reactions of adsorbed compounds, the concentration curves of the compounds in bulk phase will be qualitatively of the same form as the curves typical for noncatalytic consecutive (cf. Fig. 3b) or parallel reactions. However, anomalies in the course of bulk concentration curves may occur if the rate of one or more steps of adsorption-desorption character becomes comparable or even significantly lower then the rates of surface reactions, i.e. when surface and bulk concentration are not in equilibrium. 

We shall now assume that it is possible to have a system in equilibrium composed of the various phases at a specified temperature and total pressure. This will be characterised by certain definite relations between the compositions of the phases (for example, a solid salt, saturated solution, vapour of the solvent). Let 77, T = total pressure, and temperature, of the system. n = number of components (cf. 84). r = phases ... 

The relation between CAi[ and CAi2 is determined by the phase equilibrium relationship since the molecular layers on each side of the interface are assumed to be in equilibrium with one another. It may be noted that the ratio of the differences in concentrations is inversely proportional to the ratio of the mass transfer coefficients. If the bulk concentrations, CAt> and CA02 are fixed, the interface concentrations will adjust to values which satisfy equation 10.98. This means that, if the relative value of the coefficients changes, the interface concentrations will change too. In general, if the degree of turbulence of the fluid is increased, the effective film thicknesses will be reduced and the mass transfer coefficients will be correspondingly increased. 

These relations between the various coefficients are valid provided that the transfer rate is linearly related to the driving force and that the equilibrium relationship is a straight line. They are therefore applicable for the two-film theory, and for any instant of time for the penetration and film-penetration theories. In general, application to time-averaged coefficients obtained from the penetration and film-penetration theories is not permissible because the condition at the interface will be time-dependent unless all of the resistance lies in one of the phases. 

Phase solubility analysis is a technique to determine the purity of a substance based on a careful study of its solubility behavior [38,39]. The method has its theoretical basis in the phase mle, developed by Gibbs, in which the equilibrium existing in a system is defined by the relation between the number of coexisting phases and components. The equilibrium solubility of a material in a particular solvent, although a function of temperature and pressure, is nevertheless an intrinsic property of that material. Any deviation from the solubility exhibited by a pure sample arises from the presence of impurities and/or crystal defects, and so accurate solubility measurements can be used to deduce the purity of the sample. 

It can be assumed for almost all practical cases that equilibrium exists at the interface between the two phases. The concentrations of solute A at the interface, C A G and CAL, are related by the equilibrium relation of Eq. (5) and which, for gases in general and hydrogen in particular, is often described using simplified Henry s law applied at the interface (Eq. (6)). 

Taking the balance, around both phases, effectively disregards the rate of solute transfer from liquid to solid and, instead, the assumption of a perfect equilibrium stage is employed to provide a relation between the resulting liquid and solid phase concentrations. For the special case of a linear equilibrium... 

---