The Equality of Fugacities

4represents the famous equality ofJUgacitiesfhst we already used in Chapters 9 and 11 to provide a physical interpretation of fugacity. 

Use of the equality of these abstract fiigacities in the determination of numerical answers in phase equilibrium problems was made  

4 represents, thus, the starting point for phase equilibrium calculations as we will see in Chapters 13 and 14. 

3 was first develop by Gibbs, who provided thus the solution of the phase equilibrium problem in the abstract level (see Qiapter 4) its translation however into the real world is a very difficult process as we will see in Chapters 13 and 14. 

We consider next the implications of Eq. 12.3.9 in the case of a reacting system reaching equilibrium at constant T and P in a single phase. 

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When two liquid phases exist, the compositions of the two phases a and / are governed by the equality of fugacities for each component ... 

In the case of the flash calculations, different algorithms and schemes can be adopted the case of an isothermal, or PT flash will be considered. This term normally refers to any calculation of the amounts and compositions of the vapour and the liquid phase (V, L, y,-, xh respectively) making up a two-phase system in equilibrium at known T, P, and overall composition. In this case, one needs to satisfy relation for the equality of fugacities (eq. 2.3-1) and also the mass balance equations (based on 1 mole feed with N components of mole fraction z,) ... 

Since coexisting phases of saturated liquid and saturated vapor are in < librium, the equality of fugacities as expressed by Eqs. (11.22) and (11.24) is criterion of vapor/liquid equilibrium for pure species. 

Because of the equality of fugacities of saturated liquid and vapor, calculation of fugacity for species i as a compressed liquid is done in two ste First, one calculates the fugacity coefficient of saturated vapor = by integrated form of Eq. (11.20), evaluated for P = P . Then by Eqs. (11. and (11.23),... 

Vapor-liquid phase equilibrium calculations have to be conducted for the estimation of solubility in the vapor phase (16,17). Alternatively, a cubic EOS can be applied for the estimation of properties of the liquid phase. The equality of fugacity in the two phases can be written as... 

Besides being expressed in terms of activity coefficients, the fugacities of a liquid solution can also be calculated from equations of state in the form of a fugacity coefficient ( ) . The equality of fugacities of two liquid phases at equilibrium becomes expressed by... 

Consider an organic liquid phase (H) in e( uilibrium with an aqueous phase (W). The basic thermodynamic law governing the phase equilibrium is the equality of fugacities of all components in the two phases. 

The starting point for any phase equilibrium calculation is, of course, the equality of fugacities of each species in each phase, that is. 

In this chapter we continue the discussion of fluid phase equilibria by considering examples other than vapor-liquid equilibria. These other types of phase behavior include the solubility of a gas (a substance above its critical temperature) in a liquid, liquid-liquid and vapor-liquid-liquid equilibria, osmotic equilibria, and the distribution of a liquid solute between two liquids (the basis for liquid extraction). In each of these cases the starting point is the same the equality of fugacities of each species in all the phases in which it appears. 

In this chapter we consider several other types of phase equilibria, mostly involving a fluid and a solid. This includes the solubility of a solid in a liquid, gas, and a supercritical fluid the partitioning of a solid (or a liquid) between two partially soluble liquids the freezing point of a solid from a liquid mixture and the behavior of solid mixtures. Also considered is the environmental problem of how a chemical partitions between different parts of the environment. Although these areas of application appear to be quite different, they are connected by the same starting point as for all phase equilibrium calculations, which is the equality of fugacities of each species in each phase ... 

From the equality of fugacities, we have, for a component distributed between air and water. 

Fugacity is a key concept in phase equilibria. The phase equilibrium condition consists of the equality of fugacities of a component among coexistent phases. The computation of fugacities implies two routes equation of states, for both pure components and mixtures, and liquid activity coefficients for non-ideal liquid mixtures. The methods based on equations of state are more general. 

For any component i of a multicomponent, multiphase system, derive (4-12), the equality of fugacity, from (4-8), the equality of chemical potential, and (4-11), the definition of fugacity. 

As a particular example, consider the isotherm at 30°C. In the figure this isotherm is marked by the letters B and E on the phase equilibrium curve, where the equality of fugacities (8.4.34) is satisfied, and it is marked with C and D on the spinodal, where (8.4.36) is obeyed. Therefore at 30°C,... 

Start with the equality of fugacities (7.3.12) for vapor-Uquid equilibrium and perform the steps cited in 8.2.5 to derive (8.2.21) for pxue-component vapor pressures. Continue the derivation to obtain the equal-area form (8.2.22). 

Liquid-solid equilibria are attacked with the gamma-gamma method in the same general way as liquid-liquid systems however, the two applications differ in how the standard-state fugacities are treated. We still start from the equality of fugacities. 

Every phase-equilibrium problem is solved by starting from the phase-equilibrium conditions (7.3.12), which express the equality of fugacities. 

Phi-phi form. If we choose FFF 1 for both vapor and liquid, then the equality of fugacities takes the phi-phi form (10.1.3), and the K-factor becomes... 

As a typical example, consider the solubility of a solid in a supercritical fluid. For the equality of fugacities we choose the gamma-phi form and use FFF 5 for the condensed phase then for solute i we write... 

The equality of fugacities is an alternative statement of the necessary and sufficient condition for phase equilibrium and the basis for all such calculations, whether we are dealing with pure substances or with mixtures. By contrast, the equality of the fugacity coefficients is a special result and applies to pure substances only. 

The basic relationship for every component i in the vapor and liquid phases of a system at equilibrium is the equality of fugacities in all phases. 

Consider the equilibrium between a solid, component 2, and some gas, component 1, at some temperature T and pressure P. Since the solubility of the gas in the solid is negligible, the equality of fugacities for component 2 in the two phases yields ... 

Finally, in certain cases special constraints can be imposed on the system beyond the k 4> - 1) equations resulting from the equality of fugacities. A more general expression, therefore, of the phase rule for a nonreacting system is ... 

We have seen in Chapter 12 that when a system containing two phases reaches equilibrium - at constant temperature and pressure - the total Gibbs free energy assumes its minimum value and that this, in turn, leads to the equality of fugacities for any component i in the two phases (Section 12.4.4). In the case of vapor-liquid equilibrium, thus ... 

It is apparent that Eq. 13.5.1, obtained from the equality of fugacities, cannot provide the answer, even if we assume ideal vapor behavior. We still have ... 

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