Fluid phase equilibrium fugacity

The solubility increases with increase in pressure at a hxed temperature, owing to enhanced solvation due to greater attractive forces between the solute and carbon dioxide. A fundamental relationship for phase equilibrium (Prausnitz et al. 1999) can be used to relate fugacities of the solute in the solid and fluid phases as follows ... 

As in liquid-liquid or vapor-liquid equilibria, when a liquid or vapor is in contact with a sorbent, equilibrium is established at the solid surface between the compositions of a solute in the two phases. This is expressed in terms of the concentration of the solute in the sorbent as a function of its concentration in the fluid phase. Whereas phase equilibrium in vapor-liquid or liquid-liquid systems can be estimated based on the thermodynamic condition of equality of component fugacities in the phases, no valid theory exists for predicting solid-fluid systems. Equilibrium concentrations for these systems must be based on experimental data. 

In pioneering research by Hailing and co-workers, it was demonstrated that the activity of water is a more representative and useful parameter than water concentration for describing enzymatic rates in nonaqueous enzymology. Water activity, or is defined as the fugacity of water contained in a mixture divided by the fugacity of pure water at the mixture s temperature. For a typical nonaqueous enzymatic reaction operated in a closed system, the medium will consist of a solvent (or fluid) phase, an enzyme-contaiifing solid phase, and air headspace above the solvent. As a first approximation, the water transport between the three phases is assumed to be at thermodynamic equilibrium. For such a situation, can be defined in terms of the air headspace properties ... 

The principles and algorithms for calculating fluid-phase equilibria are discussed in many textbooks [36 0]. Here, we focus on methods and data requirements for calculating the component fugacities in a phase as a function of temperature, pressure, and composition this is the key element in all phase-equilibrium calculations. 

An equation of state, applicable to all fluid phases, is paitiodariy useful for phase-equilibrium calculations where a liquid phase and a vapor phase coexist at high pressures. At such conditions, conventional activity coefficients are not useful because, with rare exceptions, at least one of the mixture s components is supercritical that is, (he system temperature is above (hat component s critical temperature. In that event, one must employ special standard states for the activity coefficients of the supercritical components (see Section 1.5-2). That complication is avoided when ail fugacities are calculated front en equation of state. 

Unfortunately, very few mixtures are ideal gas mixtures, so general methods must be developed for estimating the thermodynamic properties of real mixtures. In the dis-, cussion of phase equilibrium in a. pure fluid of Sec. 7.4, the fugacity function was especially useful the same is true for mixtures. Therefore, in an analogous fashion to the derivation in Sec. 7.4. we start from... 

The fugacity function is central to the calculation of phase equilibrium. This should be apparent from the earlier discussion of this chapter and from the calculations of Sec. 7.5, which established that once we had the pure fluid fugacity, phase behavior in a pure fluid could be predicted. Consequently, for the remainder of this chapter we will be concerned with estimating the fugacity of species in gaseous, liquid, and solid mixtures. 

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 ... 

Evaluation of Fugacities Using an Equation of State. The fugaci-ties of the components in the fluid phases are related to the volumetric and phase behavior of the mixture while the fugacity of the solid component depends only on the PVT relationship of the pure component. Theoretically it is possible to evaluate the fugacities using experimental volumetric and/or phase equilibrium data in conjunction with Equations 3 and 6. However, these data are normally either unavailable or insufficient and an equation-of-state model has to be used to compute the fugacities. 

From the outset the relationships between the fugacity and the state variables are highly nonlinear. To determine the composition of each phase for a SLV system such that Equations 1 and 2 are satisfied requires that an iterative method be used. Because of the constraints imposed on the system by the phase rule somewhat different procedures were used in this study to compute the SLV equilibrium condition for multicomponent systems and for binary systems, respectively. Both procedures calculate the fluid-phase compositions of a given mixture at the incipient solid-formation condition. 

The criteria for equilibria involving solid phases are exactly those given in 7.3.5 for any phase-equilibrium situation phases in equilibrium have the same temperatures, pressures, and fugacities. Moreover, pure-component solid-fluid equilibria obey the Clapeyron equation (8.2.27). This means the latent heat of melting is proportional to the slope of the melting curve on a PT diagram and the latent heat of sublimation is proportional to the slope of the sublimation curve. In the case of solid-gas equilibria, the Clausius-Clapeyron equation (8.2.30) often provides a reliable relation between temperature and sublimation pressures, analogous to that for vapor-liquid equilibria. 

When two or more fluid phases are in physical equilibrium, the chemical potential, fugac-ity, or activity of each species is the same in each phase. Thus, in terms of species mixture fugacities for a vapor phase in physical equilibrium with a single liquid phase,... 

The fugacity of the pure solid compound 2 can be described by the sublimation pressure, the fugacity coefficient in the saturation state and the Poyntin.g factor, so that the following phase equilibrium relation is obtained for the calculation of the solubility of the solid 2 in the supercritical fluid ... 

In a sorption process, a species from a fluid phase binds to the surface of (in the case of adsorption) or in the interior of (for absorption) another condensed phase. Thus, one can find sorption of gas molecules to liquids, adsorption of gas molecules to the surface of liquids or solids, or sorption of solute species (ions or molecules) from solution to solid particles. An adsorption isotherm or, more generally, a sorption isotherm, is an equilibrium relationship between the activity (or fugacity) of the species to be bound, the sorbate, in the bulk phase and the activity of the bound sorbate. The process can be regarded formally as a chemical reaction, for example, in the case of adsorption from aqueous solution... 

Extreme care must be exercised in choosing a model for phase equilibria (sometimes called the fugacity coefficient, K-factor, or fluid model). Whenever possible, phase equilibrium data for the system should be used to regress the parameters in the model, and the deviation between the model predictions and the experimental data should be studied. 

One of the simplest cases of phase behavior modeling is that of soHd—fluid equilibria for crystalline soHds, in which the solubility of the fluid in the sohd phase is negligible. Thermodynamic models are based on the principle that the fugacities (escaping tendencies) of component are equal for all phases at equilibrium under constant temperature and pressure (51). The soHd-phase fugacity,, can be represented by the following expression at temperature T ... 

The equation, referred as the SRK eos, gives quantitative fitting of vapor pressure, good representation of the fugacity of liquid, and improved representation of the energy functions of liquid for normal fluids, although liquid density is not well represented. The equation was the first to be widely used for both the gas and liquid phases and, hence, for gas-liquid equilibrium in engineering calculations. 

In en equilibrated, multiphase mixture at temperature Tand pressure P, foreveiy component r, (he fugacity /, must be the same In all phases. Therefore, if a method is available for calculating /, for phase as a function of temperature, pressure, and the phase s composition, (hen it is possible to calculate ail equilibrium ocxnpositjoua at any desired temperature and pressure. When all phases are fluids, such a method is given by an equation of state. 

We now want to consider the extent to which a solid is soluble in a liquid, a gas. or a supercritical fluid. (This last case is of interest for supercritical extraction, a new separation method.) To analyze these phenomena we again start with the equality of the species fugacities in each phase. However, since the fluid (either liquid, gas, or supercritical fluid) is not present in the solid, two simplifications arise. First, the equilibrium criterion applies only to the solid solute, which we denote by the subscript 1 and second, the solid phase fugacity of the solute is that of the pure solid. Thus we have the single equilibrium relation... 

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