Equilibrium expression fugacity

The method of using fugacity calculations will be discussed later in this symposium, therefore a detailed description will not be given in this paper. The description of equilibrium models using chemical equilibrium expressions will be discussed with the recognition that the two approaches are very much the same. 

Since chemical equilibrium expressions involve fugacity coefficients, rather than K-values, it is necessary to introduce a new set of outside loop variables. The K-value is factored as follows ... 

A more rigorous expression is derived by noting that at equilibrium, partial fugacities of each component are the same in each phase, that is... 

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 equilibrium expression can be formed from any concentration measure and a suffix on A is a convenient way of making the distinction from K with no suffix which is calculated from fugacities. Thus Eq. (3.6.10) might be written... 

These are C equations, corresponding to the C components. The distribution coefficients are functions of the temperature, pressure, and liquid and vapor compositions. Equation 2.9 is based on the thermodynamic condition that at phase equilibrium the fugacities of each component in the liquid phase and the vapor phase are equal. Referring to Section 1.3, the fugacities can be expressed in terms of fugacity coefficients as follows ... 

Now, the problem is to express fugacity as function of P, T and composition, and on this basis to determine the equilibrium composition. Two possibilities exist assume ideal or real solutions. 

To form an equilibrium ratio, fugacities are replaced by equivalent expressions involving mole fractions. Many replacements are possible. Two common pairs derived from (4-16) through (4-20) are Pair 1 ... 

Substituting this equilibrium expression for the anode fugacity and assuming an ideal gas yields ... 

The fugacity bears the same relationship to the chemical potential for real fluids as does the partial pressure for ideal gases. Because of the direct relationship between chemical potential and fugacity, the condition for equilibrium expressed by Equation 1.9 is equivalent to the equality of component fugacities in the phases ... 

T aking the concept of fugacity into account, these equilibrium expressions can thus be written as... 

The most common phase equilibria problems chemical engineers encounter involve vapor-liquid equilibrium (VLE). We can write the general expression for VLE by applying the defining relations in Chapter 7. At equilibrium, the fugacity of species i in the vapor and liquid are equal ... 

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 ... 

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 ... 

In apphcatious to equilibrium calculations, the fugacity coefficients of species iu a mixture are required. Given au expression for G /RT as aetermiued from Eq. (4-158) for a coustaut-compositiou mixture, the corresponding recipe for In is found through the partial-property relation... 

The phase-distribution restrictions reflect the requirement that ff =ff at equilibrium where/is the fugacity. This may be expressed by Eq. (13-1). In vapor-hquid systems, it should always be recognized that all components appear in both phases to some extent and there will be such a restriction for each component in the system. In vapor-liquid-hquid systems, each component will have three such restrictions, but only two are independent. In general, when all components exist in all phases, the uumDer of restricting relationships due to the distribution phenomenon will be C(Np — 1), where Np is the number of phases present. 

Thus, equilibrium is achieved when the escaping tendency from the vapor and liquid phases for Component i are equal. The vapor-phase fugacity coefficient, fj, can be defined by the expression ... 

Equilibrium. Equilibrium between compartments can be expressed either as partition coefficients K.. (i.e. concentration ratio at equilibrium) or in the fugacity models as fugacity capacities and Z. such that K.. is Z./Z., the relationships being depicted in Figur 1. Z is dellned as tfte ratio of concentration C (mol/m3) to fugacity f (Pa), definitions being given in Table I. 

The feed stream consists of 60 mole percent hydrogen, 20% nitrogen, and 20% argon. Calculate the composition of the exit gases, assuming equilibrium is achieved in the reactor. Make sure that you take deviations from the ideal gas law into account. The equilibrium constant expressed in terms of activities relative to standard states at 1 atm may be assumed to be equal to 8.75 x 10 3. The fugacity of pure H2 at 450 °C and 101.3 MPa may be assumed to be equal to 136.8 MPa. 

To this point we have assumed the existence of a basis of chemical components that corresponds to the system to be modeled. The basis, as discussed in the previous chapter, includes water, each mineral in the equilibrium system, each gas at known fugacity, and certain aqueous species. The basis serves two purposes each chemical reaction considered in the model is written in terms of the members of the basis set, and the system s bulk composition is expressed in terms of the components in the basis. 

The procedure of Beutier and Renon as well as the later on described method of Edwards, Maurer, Newman and Prausnitz ( 3) is an extension of an earlier work by Edwards, Newman and Prausnitz ( ). Beutier and Renon restrict their procedure to ternary systems NH3-CO2-H2O, NH3-H2S-H2O and NH3-S02 H20 but it may be expected that it is also useful for the complete multisolute system built up with these substances. The concentration range should be limited to mole fractions of water xw 0.7 a temperature range from 0 to 100 °C is recommended. Equilibrium constants for chemical reactions 1 to 9 are taken from literature (cf. Appendix II). Henry s constants are assumed to be independent of pressure numerical values were determined from solubility data of pure gaseous electrolytes in water (cf. Appendix II). The vapor phase is considered to behave like an ideal gas. The fugacity of pure water is replaced by the vapor pressure. For any molecular or ionic species i, except for water, the activity is expressed on the scale of molality m ... 

Vapor-liquid equilibrium data for the two binary systems (11) were used to calculate the standard-state fugacities required in Equations 6 and 20. In the temperature range 0-50°C, there fugacities can be expressed by ... 

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 ... 

The dashed line in Fig. 19.7 gives the concentration in zone B expressed as the corresponding A-phase equilibrium concentration. This modified representation is like an extrapolation of the A-phase concentration scheme into system B. In fact, it is the same as considering the variability of activity or fugacity of the chemical, rather than its concentration, through the adjacent media. Consequently, the concentration jump at the phase boundary disappears the concentration profile (or more accurately the chemical activity profile) across the boundary looks like that shown in Fig. 19.6. 

Equilibrium in an accumulation process is empirically defined as the point at which a partitioning coefficient becomes invariant (reaches steady state), and theoretically as the point at which the fugacity ratio equals 1 (21). With phytoplankton, most published reports indicated that equilibrium was reached in a matter of hours (6, 8, 22-26). As a result, predictions of HOC accumulation in phytoplankton have been expressed as equilibrium-based equations exclusively. 

Eq 18 is applicable to equilibrium constants in alternate form. For ideal gases the equilibrium constant Kp is expressed in terms of partial pressures (rather than fugacities) of products and reactants. Still another form of the equilibrium constant. JC. is exnressed in terms nf... 

Solubility normally is measured by bringing an excess amount of a pure chemical phase into contact with water at a specified temperature, so that equilibrium is achieved and the aqueous phase concentration reaches a maximum value. It follows that the fugacities or partial pressures exerted by the chemical in these phases are equal. Assuming that the pure chemical phase properties are unaffected by water, the pure phase will exert its vapor pressure Ps (Pa) corresponding to the prevailing temperature. The superscript S denotes saturation. In the aqueous phase, the fugacity can be expressed using Raoult s Law with an activity coefficient y ... 

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