Fluid phase fugacities

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 fugacity coefficient of thesolid solute dissolved in the fluid phase (0 ) has been obtained using cubic equations of state (52) and statistical mechanical perturbation theory (53). The enhancement factor, E, shown as the quantity ia brackets ia equation 2, is defined as the real solubiUty divided by the solubihty ia an ideal gas. The solubiUty ia an ideal gas is simply the vapor pressure of the sohd over the pressure. Enhancement factors of 10 are common for supercritical systems. Notable exceptions such as the squalane—carbon dioxide system may have enhancement factors greater than 10. Solubihty data can be reduced to a simple form by plotting the logarithm of the enhancement factor vs density, resulting ia a fairly linear relationship (52). 

For more comprehensive calculations of the fugacity coefficients in mixtures, see J. M. Prausnitz, R. N. Lichtenthaler, and E. G. de Azevedo. Modular Thermodynamics of Fluid Phase Equilibria, Prentice Hall. Englewood Cliffs. N.J., 19S6. Chapter 5. 

In 1977 De Santis et al. (J5) as well as Heidemann et al. ( ) calculated the gas-phase fugacities in the systems HjO-air and H2O-N2-CO2 by equation of state in these calculations the liquid phase was not included. One of the authors (7J showed in 1978 that aqueous systems with some inert gases and alkanes as well as H2S and C02 could be represented by an equation of state if the molecular weight of water was artificially increased. An extension of this method applied to alcohols was found to be only partially successful. Gmehling et al. (8) treated polar fluids such as alcohols, ketones and water as monomer-dimer mixtures using Donohue s equation of state (9) various systems including water-methanol and water-ethanol were succussfully represented. 

Jager, M.D. Ballard, A.L. Sloan, E.D. Jr. (2003). The next generation of hydrate prediction - II. Dedicated aqueous phase fugacity model for hydrate prediction. Fluid Phase Equilibria, 211, 85-107. 

According to eq. (2.3-30), the temperature mainly influences the sublimation pressure and the fugacity coefficient in the supercritical phase. The sublimation pressure always increases with increasing temperature, and if this is the main effect the solubility must always increases with temperature. To the contrary, at relatively low pressures (close to the critical pressure of the supercritical fluid) the fugacity coefficient in the supercritical phase plays the most important and preponderant role. 

This equation is analogous to Eq. 5 of Ch. 1 for the solubility of a solid in a SCF. In this equation, the subscript 2 refers to the liquid component. The superscript s refers to saturation conditions at temperature T. Pj refers to the saturation vapor pressure of the liquid at temperature T. The variable uf is the molar volume of the liquid, ( )2 is the fugacity coefficient at saturation pressure and is the fiigacity coefficient in the high pressure gas mbrture. For a detailed derivation of this equation, see Prausnitz. " As is stated in the derivation, it is the escaping tendency of the liquid into the supercritical fluid phase, as described by the fugacity coefficient, ( >2, which is responsible for the enhanced solubility of liquids in compressed gases. 

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 ... 

The fugacity of the solute in fluid phase (02) can be calculated from (Prausnitz et al. 1999) ... 

Equation 7-14 is used to calculate the reference state fugacity of liquids. Any equation of state can be used to evaluate ([) . For low to moderate pressures, the virial equation is the simplest to use. The fugacities of pure gases and gas mixtures are needed for estimating many thermodynamic properties, such as entropy, enthalpy, and fluid phase equilibria. For pure gases, the fugacity is... 

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. 

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. 

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... 

All of these fugacities are evaluated at the temperature and pressure of the system. The fugacities of component k for the fluid phases can be calculated using the rigorous thermodynamic equation... 

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. 

Equation 20 shows that the fugacity coefficient in the liquid, iL, when properly evaluated, provides a self-sufficient thermodynamic quantity in the analysis of high-pressure fluid-phase equilibria. 

The numerator quantifies the effect of hydrostatic pressure on the fugacity of the solid phase. The exponential term is known as the Poynting correction (17). The denominator quantifies the fluid phase intermolecular interactions and density effects. Note that the enhancement factor is dependent on the solid volume as well as the interactions in the supercritical fluid. A solute with a large solid molar volume will have a larger enhancement factor than a solute with a smaller solid molar volume at the same temperature and pressure when the interactions in the supercritical phase are identical. To further understand the molecular interactions in supercritical fluids, it is interesting to decompose the enhancement factor into these two effects. We may define a fluid enhancement factor, Ep, and a Poynting enhancement factor, Ep,... 

The phase behavior and solubility of soHds in supercritical fluids is quite different than that in liquids. First, it is assumed that the solubility of the fluid-phase component, such as supercritical CO2, in the solid phase is negligible. This is unlike the equihbrirmi with liquids, where one must consider the mutual solubilities of both the liquid solute and the fluid phase. From thermodynamics, the mole fraction solubility of a solute in a supercritical fluid, y2, is given by Eq. (4), where, P is the sublimation pressure of the solid which is a function of temperature alone, the exponential term is called the Poynting correction (usual values are 1 to 4) to account for hydrostatic pressure, and the fugacity coefficient, ]>, which accounts for the non-idealities of the fluid phase at a certain temperature, pressure, and concentration. At low pressures, the behavior is ideal and the solubility y2 is equal to P /P. 

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,... 

Plyasunov, A. V. 2011. Thermodynamics of B(OH)3 in the vapor phase of water Vapor-liquid and Henry s constants, fugacity and second cross virial coefficients. Fluid Phase... 

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