Liquid, fugacity general equation

These are general equations that do not depend on the particular mixing rules adopted for the composition dependence of a and b. The mixing rules given by Eqs. (4-221) and (4-222) can certainly be employed with these equations. However, for purposes of vapor/liquid equilibrium calculations, a special pair of mixing rules is far more appropriate, and will be introduced when these calculations are treated. Solution of Eq. (4-232) for fugacity coefficient at given T and P reqmres prior solution of Eq. (4-231) for V, from which is found Z = PV/RT. 

Effective use of this general equation requires explicit introduction of the compositions of the phases. This is done either through the activity coefficient, y, or the fugacity coefficient, ( ) A Two procedures are in common use. By the gamma—phi approach, activity coefficients for the liquid phase enter by equation 202 and fugacity coefficients for the vapor phase by equation 164 equation 220 then becomes equation 221 ... 

The Chao-Seader and the Grayson-Streed methods are very similar in that they both use the same mathematical models for each phase. For the vapor, the Redlich-Kwong equation of state is used. This two-parameter generalized pressure-volume-temperature (P-V-T) expression is very convenient because only the critical constants of the mixture components are required for applications. For the liquid phase, both methods used the regular solution theory of Scatchard and Hildebrand (26) for the activity coefficient plus an empirical relationship for the reference liquid fugacity coefficient. Chao-Seader and Grayson-Streed derived different constants for these two liquid equations, however. 

The exponential term in this equation, known as the Poynting pressure correction, accounts for the increase in fugacity due to the fact that the system pressure is higher than the vapor pressure of the liquid. Since the molar volmne of a liquid is generally much less than that of a gas (so that PV /RT -C 1), the Poynting term is only important at high pressures. (An exception to this is for cryogenic systems, where T is very low.)... 

A new pressure-explicit equation of state suitable for calculating gas and liquid properties of nonpolar compounds was proposed. In its development, the conditions at the critical point and the Maxwell relationship at saturation were met, and PVT data of carbon dioxide and Pitzers table were used as guides for evaluating the values of the parameters. Furthermore, the parameters were generalized. Therefore, for pure compounds, only Tc, Pc, and o> were required for the calculation. The proposed equation successfully predicted the compressibility factors, the liquid fugacity coefficients, and the enthalpy departures for several arbitrarily chosen pure compounds. 

Eq. (1.40) is a general equation for the solubility of any solute in any solvent. We can see from this equation that the solubility depends on the activity coefficient and on the fugacity ratio filft- The standard state fugacity normally used for solid-liquid equilibrium is the fugacity of the pure solute in a subcooled liquid state below its melting point. We can simplify Eq. (1.40) further by assuming that our solid and subcooled liquid have small vapor pressures. We can then substitute vapor pressure for fugacity. If we further assume that the solute and solvent are chemically similar so that 72 = 1, then we can write... 

The knowledge of equations of state for gas phases permits the calculation of activity coefficients via fugacity coefficients. Equations of state for general practical use such as the virial equation (and others) are not known for condensed phases (liquids and solids). However, as shown by Planck and Schottky, the passage from the gaseous to the liquid or solid state does not change the structure of Eq. (87) and leads to the general formulation for the chemical potentials,... 

There are two general types of fugacity models equations of state and liquid-state activity-coefficient models. An equation of state is an algebraic equation for the pressure of a mixture as a function of the composition, volume, and temperature. Through standard thermodynamic relationships, the fugacity, enthalpy, and so on for the mixture can be determined. These properties can be calculated for any density therefore, both liquid and vapor properties, as well as supercritical phenomena, can be determined. 

The separation of the vapor and liquid fugacities and the activity coefficients in the fundamental equilibrium relationship allow great flexibility, and a multitude of choices, in the selection of the thermodynamic relationships or empirical equations for estimation of each of these quantities. For the vapor fugacity coefficient any of the equations of state mentioned earlier or some other, such as the virial equation, may be used. In the latter case, the virial coefficients may be determined experimentally or estimated using three- or four-parameter generalized correlations. 

A question of some importance is Does the curved surface of a drop or other geometries affect the chemical potential (or fugacity) of a species compared with that on a planar surface For gas-liquid, vapor-liquid, solid-liquid or liquid-liquid systems, this subject has been considered by Lewis and Randall (1961), starting from the general equation of a differential change in total Gibbs free energy of a phase j (compare with relation (3.3.2)) ... 

The aqueous solubility of a gaseous compound is commonly reported for 1 bar (or 1 atm = 1.013 bar) partial pressure of the pure compound. One of the few exceptions is the solubility of 02 which is generally given for equilibrium with the gas at 0.21 bar, since this value is appropriate for the earth s atmosphere at sea level. As discussed in Chapter 3, the partial pressure of a compound in the gas phase (ideal gas) at equilibrium above a liquid solution is identical to the fugacity of the compound in the solution (see Fig. 3.9d). Therefore equating fugacity expressions for a compound in both the gas phase and an equilibrated aqueous solution phase, we have ... 

Equations of state have a much wider application. In this chapter we first present a general treatment of the calculation of thermodynamic properties of fluids and fluid mixtures from equations of state. Then the use of an equation of state for VLE calculations is described. For this, the fugacity of each species in both liquid and vapor phases must be determined. These calculations are illustrated with the Redlich/Kwong equation. Provided that the equation of state is suitable, such calculations can extend to high pressures. 

Equations (6.1) and (6.2) pertain to ideal systems, that is, systems where there are no interactions between the molecules. In a real system the pressure effect on p. in the vapor phase has to be modified by a fugacity coefficient < >, and the effect of mixing on the chemical potential in the liquid phase has to be modified by an activity coefficient 7,. The more general expression for equilibrium (called the 4>-y representation) then becomes... 

A generally applicable alternative to the gamma/phi approach results when both the liquid and vapor phases are described by the same equation of state. The defining equation for the fugacity coefficient, Eq. (4-79), may be applied to each phase ... 

The description of vapour-liquid equilibrium behaviour can be obtained from analytical equations and generalized correlations. The generalized conelations are generally for the equilibrium ratio, K, and the fugacity coefficients. 

An equation, somewhat similar to (35.6), was suggested by M. Margules (1895) to express the variation bf vapor pressure with composition of liquid mixtures in general replacing the vapor pressure by the fugacity, this can be written as... 

By equation (31.5), the activity of the solvent is equivalent to fi/fi where fi is the fugacity in a given solution and / is numerically equal to that in the standard state, i.e., pure liquid at 1 atm. pressure at the given temperature. Hence, it is seen from equation (34.1) that for an ideal solution the activity of the solvent should always be equal to its mole fraction, provided the total pressure is 1 e m. In other words, in these circumstances the activity coefficient ui/ni should be inity at all concentrations. For a nonideal solution, therefore, the deviation of ai/Ni from unity at 1 atm. pressure may be taken as a measure of the departure from ideal (Raoult law) behavior. Since the activities of liquids are not greatly affected by pressure, this conclusion may be accepted as generally applicable, provided the pressure is not too high. 

To use this equation we must estimate the pure component fugacity of each species as both a liquid and a vapor at the temperature and pressure of the mixture. However, at this temperature and pressure either the liquid or the vapor will be the stable phase for each species, generally not both. 

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. 

Ideal A/B mixtures that can be described by Raoult s law do not exist in practice. Generally, mixtures that have a more or less pronounced real behavior are encountered. The deviations from ideal behavior are taken into account by correcting factors known as activity coefficients y in the case of the liquid phase or fugacity coefficients 0 in the case of the gas phase (Equation 2.3.2-10 see also Section 3.3.2) ... 

At high pressures, the rhs of (10.1.4) should be replaced by one of FFF 3-5. Which of those to choose depends on what other data are available for the component general considerations have been discussed in 6.4. When FFF 3 or 5 is chosen, a Po)mt-ing factor must be evaluated, often using a volumetric equation of state for the liquid that equation of state need not be the same as the one used for obtaining the fugacity coefficients for the vapor (phase a). 

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