Fugacity of a solid

In Equation 10.22 through Equation 10.24, X2, x, and X2 are the solubilities (mole fractions) of the solid (component 2) in the cosolvent, water, and their mixture, respectively, Y > and Y2 are the activity coefficients of the solid in its saturated solutions in cosolvent, water, and mixed solvent,/2 (T,p) is the hypothetical fugacity of a solid as a (subcooled) liquid at a given pressure and temperature,/2 is the fugacity of the pure solid (component 2), and x designates that the activity coefficients of the solid depend on composition. If the solubilities of the pure and mixed solvents in the solid phase are negligible, then the left-hand sides of Equation 10.22 through Equation 10.24 depend only on the properties of the solute. 

Because is adimensional, it is evident that fugacity has the dimension of pressure. It is important to emphasize that it is conceptually erroneous to consider the fugacity of a gaseous species as the equivalent of the thermodynamic activity of a solid or liquid component (as is often observed in literature). The relationship between activity a, and fugacity f of a gaseous component is given by... 

In this approach the standard state fugacity of a liquid or solid component is usually the fugacity of die pure solid or liquid component, and is closely related to the sublimation pressure P- b or vapour pressure P.at, respectively. I.e., on the sublimation curve of a pure component we have... 

When a drop of liquid is placed on the surface of a solid, it may spread to cover the entire surface, or it may remain as a stable drop on the solid. There is a solid-liquid interface between the two phases. In the case of liquids that do not spread on the solid, the bare surface of the solid adsorbs the vapor of the liquid until the fugacity of the adsorbed material is equal to that of the vapor and the liquid. 

Horstmann, M. and M.S. McLachlan. 1992. Initial Development of a Solid-Phase Fugacity Meter for Semivolatile Organic Compounds. Environ. Sci. Technol. 26, 1643-1649. 

The fugacity of a pure liquid or solid can be defined by applying Eq. si.4 to the vapor in equilibrium with the substance in either condensed phase. Usually, the volume of the vapor will follow the ideal gas equation of state very closely, and the fugacity of the vapor may be set equal to the equilibrium vapor pressure. The thermodynamic basis of associating the fugacity of a condensed... 

Instead of a solid-water equilibrium relationship, a solid-air equilibrium relationship could be used in the same manner to define the fugacity capacity and the fugacity. 

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. 

Recently, a method [5] for the prediction of the solubility of a solute in a SC fluid in the presence of an entrainer has been proposed. The method, based on the Kirkwood-Buff (KB) formalism, was however developed for cases in which the entrainer was in dilute amounts. The present paper is focused on the solubility of a solid in a non-dilute mixture of a SC fluid and an entrainer. The theoretical treatment, which is more complex than for the dilute case, is also based on the KB formalism. In this paper the following aspects will be addressed (1) general equations for the solubility in binary and ternary mixtures will be written for the cases involving a small amount of solute (2) the KB formalism will be used to obtain expressions for the derivatives of the fugacity coefficients in a ternary mixture with respect to mole fractions (3) these expressions will be employed to derive an equation for the solubility of a solute in a SC fluid containing an entrainer at any concentration (4) a predictive method for this solubility will be proposed in terms of the solubilities of the solute in the SC fluid and in the entrainer (5) the derived equation will be compared with experimental results from literature regarding the solubility of a solute in a mixture of two SC fluids. 

Expression for the Fugacity Coefficient of a Solid in a Ternary Mixture at Infinite Dilution by the Kirkwood—Buff Formalism. The following... 

Equation 14 allows one to calculate the fugacity coefficient of a solute at infinite dilution in the binary mixture of two SC fluids, in terms of the fugacity coefficients of the solute at infinite dilution for each of the SC fluids. This expression will be used in the next section to derive an expression for the solubility of a solid in a gaseous mixture of two SC fluids. 

Equation 1 shows that the solubility of a solid in SCF depends among others on the fugacity coefficient large values of the solubility. These solubilities are much larger than those in ideal gases, and enhancement factors of 10 —10 are not uncommon. They are, however, still relatively small and usually do not exceed several mole percent. 

Two limitations are involved in the derivation of the above equation (1) the compositions of mixed solvents (points c and d) should be close enough to each other for the trapezoidal mle used to integrate the Gibbs-Duhem equation to be valid, (2) the solubility of the solid should be low enough for the activity coefficients of the solvent and cosolvent to be taken equal to those in a solute-free binary solvent mixture. In addition, the fugacity of the solid phase in Eq. (4) should remain the same for all mixed solvent compositions considered. 

The fate of chemicals in the environment depends not only on processes taking place within compartments, but also by chemical partitioning between compartments. For example, there may be exchange of chemicals between air and water or soil. Movement from the water or soil into the air is accomplished by volatilization and evaporation of volatile or semivolatile compounds. Movement of chemicals from the air to water or soil is accomplished by deposition or diffusion into the water. Chemicals can also move from water to soil or sediment and vice versa. If a solid chemical in the soil or sediment dissolves into the water, this is called dissolution , while the opposite is called precipitation . If a chemical dissolved in water attaches to a soil or sediment particle, this is called adsorption , while the opposite is called desorption . The fugacity of a chemical, that is, its tendency to remain within a compartment, is affected by the properties of that chemical, as well as the chemical and physical properties of the environments such as temperature, pFF, and amount of oxygen in water and soil. Wind or water currents, wave action, water turbulence, or disturbance of soil or sediment (through the action of air or water currents, animals, or human activities) may also affect partitioning of chemicals. 

The equations derived in 30c, 30d thus also give the variation with pressure and temperature of the fugacity of a constituent of a liquid (or solid) solution. In equation (30.17), Vi is now the partial molar volume of the particular constituent in the solution, and in (30.21), i is the corresponding partial molar heat content. The numerator — fti thus represents the change in heat content, per mole, when the constituent is vaporized from the solution into a vacuum (cf. 29g), and so it is the ideal" heat of vaporization of the constituent i from the given solution, at the specified temperature and total pressure. 

Experimental information on gas hydrate nucleation at a microscopic level is almost nonexistent or at best very limited. Most of the studies on the hydrate nucleation are based on a macroscopic approach. Although there are some differences, gas hydrate nucleation has similarities with salt crystal nucleation. For the nucleation to occur, supersaturation of the aqueous solution with the hydrate former gas is required. The supersaturation is necessary to overcome the free energy barrier for creating a new surface of a solid hydrate nucleus. The degree of supersaturation or the driving force for nucleation may be defined in terms of difference in the chemical potential or the fugacity of a hydrate former in the solution and that at the... 

One complication with this description is that a species can be present in a liquid mixture, though at the temperature and pressure of the mixture the substance would be a vapor or a solid as a pure component. This is especially troublesome if the compound is below its melting point, so that it is the solid sublimation pressure rather than the vapor pressure that is known, or if the compound is above its critical temperature, so that the vapor pressure is undefined. In the first case one frequently ignores the phase change and extrapolates the liquid vapor pressure from higher temperatures down to the temperature of interest using, for example, the Antoine equation, eqn. (2.3.11). For supercritical components it is best to use an EOS and compute the fugacity of a species in a mixture, as described in Section 2.5. 

In modeling mixture gas-solid equihbrium, consideration is given to pure sohd phase or phases solid solutions are so rare they will not be considered. The modeling is simplified in two ways First, modeling the fugacity of a sohd is simple second, there are no equilibrium equations for the components that do not form sofids they remain in the gas phase. Denoting a solid-forming component by i, the equihbrium equation is... 

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