Fugacity single

Thermodynamics of Liquid—Liquid Equilibrium. Phase splitting of a Hquid mixture into two Hquid phases (I and II) occurs when a single hquid phase is thermodynamically unstable. The equiUbrium condition of equal fugacities (and chemical potentials) for each component in the two phases allows the fugacitiesy andy in phases I and II to be equated and expressed as ... 

With a suitable equation of state, all the fugacities in each phase can be found from Eq. (6), and the equation of state itself is substituted into the equilibrium relations Eq. (67) and (68). For an A-component system, it is then necessary to solve simultaneously N + 2 equations of equilibrium. While this is a formidable calculation even for small values of N, modern computers have made such calculations a realistic possibility. The major difficulty of this procedure lies not in computational problems, but in our inability to write for mixtures a single equation of state which remains accurate over a density range that includes the liquid phase. As a result, phase-equilibrium calculations based exclusively on equations of state do not appear promising for high-pressure phase equilibria, except perhaps for certain restricted mixtures consisting of chemically similar components. 

So far we have defined fugacity for a single component gas. We will first see how fugacities are determined for a pure gas before we expand to include... 

FIGURE 32.4 (See color insert following page 302.) The localized minima as obtained after the GCMS simulation with carbon dioxide adsorption over single-wall CNT with a fixed fugacity of 100 kPa. 

The integration of Equation (10.83) for a component of a mixture leads to a problem of nonconvergence at P = 0, just as for a single gas. To circumvent this difficulty, we shall consider the ratio of the fugacity to the partial pressure of a component, just as we considered the ratio of the fugacity to the pressure of a single gas. 

These considerations can be extended to reversible processes. They also apply to single phase, liquid systems. For the case, rather common in heterogeneous catalysts, in which one reactant is in a gas phase and the others and the products are in a liquid phase, application of the principles given above is straightforward provided that there is mass transfer equilibrium between gas phase and liquid phase, i.e., the fugacity of the reactant in the gas phase is identical with its fugacity in the liquid phase. In such case, a power rate law for an irreversible reaction of the form... 

There are several characteristics common to the describing equations of all types of multicomponent, vapor-liquid separation processes, both single- and multi-stage, that make it possible to exploit the inside-out concept in similar ways to solve them efficiently and reliably. The equations have as common members component and total mass balance, enthalpy balance, constitutive and phase equilibrium equations. In addition, all such processes require K-value or fugacity coefficient and vapor and liquid enthalpy models. In all cases the describing equations have similar forms, and depend on the primitive variables (temperature, pressure, phase rate and composition) in essentially the same ways. Before presenting the inside-out concept, it will be useful to identify two classes of conventional methods and discuss their main characteristics. 

Two example tables are given from the set for the dissociation of water substance. These are in contrast to the single table for carbon dioxide which, being a perfect gas at relevant temperatures, contains data independent of pressure, not requiring fugacity in its analysis. 

The fugacity equations are solved using absolute variables. As discussed previously, the single-gas isotherms at sub-atmospheric pressure provide the absolute isotherm in the form fiirif). Given the temperature of the isotherms, the independent variables are P and yi in the bulk gas. For a binary mixture the fugacity equations are written ... 

Conversely, the correct approach to formulate the diffusion of a single component in a zeolite membrane is to use the MaxweU-Stefan (M-S) framework for diffusion in a nonideal binary fluid mixture made up of species 1 and 2 where 1 and 2 stands for the gas and the zeohtic material, respectively. In the M-S theory it is recognized that to effect relative motions between the species 1 and 2 in a fluid mixture, a force must be exerted on each species. This driving force is the chemical potential gradient, determined at constant temperature and pressure conditions [68]. The M-S diffiisivity depends on coverage and fugacity, and, therefore, is referred to as the corrected diffiisivity because the coefficient is corrected by a thermodynamic correction factor, which can be determined from the sorption isotherm. 

General Expressions for the Solubility of a Gas Mixture in a Single Solvent. Let us consider the solubility of a mixed gas (composed of two supercritical gases component 2 with mole fraction and component 3 with mole fraction ys) in a single solvent (component 1). At equilibrium, the fugacities of the components in the liquid and gaseous phases should be equal. Therefore, one can write... 

The arguments of the preceding paragraph apply without alteration to the case of a mixture. Thus, just as for a single gas, the fugacity may be calculated from (c/. 11.59). 

By considering the influence of temperature and pressure on the fugacities, which must remain the same in both phases, derive a form of the Clapeyron equation (27.4) for the equilibrium between two phases of a single substance. 

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