Fugacity phase equilibrium

The other state variables are the fugacity of dissolved methane in the bulk of the liquid water phase (fb) and the zero, first and second moment of the particle size distribution (p0, Pi, l )- The initial value for the fugacity, fb° is equal to the three phase equilibrium fugacity feq. The initial number of particles, p , or nuclei initially formed was calculated from a mass balance of the amount of gas consumed at the turbidity point. The explanation of the other variables and parameters as well as the initial conditions are described in detail in the reference. The equations are given to illustrate the nature of this parameter estimation problem with five ODEs, one kinetic parameter (K ) and only one measured state variable. 

The amounts of each phase and their compositions are calculated by resolving the equations of phase equilibrium and material balance for each component. For this, the partial fugacities of each constituent are determined ... 

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. 

If we could prevent the mixture from separating into two phases at temperatures below Tc, we would expect the point of inflection to develop into curves similar to those shown in Figure 8.17 as the dotted line for /2, with a maximum and minimum in the fugacity curve. This behavior would require that the fugacity of a component decreases with increasing mole fraction. In reality, this does not happen, except for the possibility of a small amount of supersaturation that may persist briefly. Instead, the mixture separates into two phases. These phases are in equilibrium so that the chemical potential and vapor fugacity of each component is the same in both phases, That is, if we represent the phase equilibrium as... 

In the case of vapor-liquid equilibrium, the vapor and liquid fugacities are equal for all components at the same temperature and pressure, but how can this solution be found In any phase equilibrium calculation, some of the conditions will be fixed. For example, the temperature, pressure and overall composition might be fixed. The task is to find values for the unknown conditions that satisfy the equilibrium relationships. However, this cannot be achieved directly. First, values of the unknown variables must be guessed and checked to see if the equilibrium relationships are satisfied. If not, then the estimates must be modified in the light of the discrepancy in the equilibrium, and iteration continued until the estimates of the unknown variables satisfy the requirements of equilibrium. 

Use of equation 247 for actual calculations requires explicit introduction of composition variables. As in phase-equilibrium calculations, this is normally done for gas phases through the fugacity coefficient and for liquid phases through the activity coefficient. Thus, either... 

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. 

Perhaps the most significant of the partial molar properties, because of its application to equilibrium thermodynamics, is the chemical potential, i. This fundamental property, and related properties such as fugacity and activity, are essential to mathematical solutions of phase equilibrium problems. The natural logarithm of the liquid-phase activity coefficient, lny, is also defined as a partial molar quantity. For liquid mixtures, the activity coefficient, y describes nonideal liquid-phase behavior. 

For condensed phases, it is the fugacity in the equilibrium vapor phase (vapor fugacity or very nearly vapor pressure) that gives the fugacity of the condensed phase. Equation (11.39) applies to relate this vapor fugacity to the chemical potential in the condensed phase. 

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 ... 

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. 

Thermodynamic properties (i.e., fugacities, entropies, and enthalpies) are required by this simulating program in the calculations of vapor/liquid phase equilibrium, compression/ expansion paths, and heat balances. Fugacities are required for the individual components of the existing vapor and liquid mixtures. Enthalpies and entropies are required for the vapor mixture or the liquid mixture. Also, mixture densities are required for both phases. 

Either set contains (tr - 1)(N) independent phase-equilibrium equations. The are equations connecting the phase-rule variables, because the chemical potenti and fugacities are functions of temperature, pressure, and composition. T difference between the number of phase-rule variables and the number o equations connecting them is the degrees of freedom ... 

Calculate the fugacity of liquid hydrogen chloride at 40°F (277.4 K) and 200 psia (1379 kPa). (The role of fugacity in phase equilibrium is discussed under Related Calculations in Example 3.1). 

To avoid some possible difficulties in determining chemical potentials, Lewis proposed a new property called the fugacity /. At low pressure and concentration, the fugacity is a well-behaved function. The fugacity function can define phase equilibrium and chemical equilibrium. For an ideal gas, the fugacity of a species in an ideal gas mixture is equal to its partial pressure. As the pressure decreases to zero, pure substances or mixtures of species approach an ideal state, and we have... 

For equilibrium between phases the fugacities for each component must be equal in the several phases. If the fugacity coefficient fa for any component i in a mixture is defined as the ratio of the fugacity fi over the vapor composition Y and the total pressure P, then... 

The general condition for phase equilibrium is the equality of chemical potentials. For practical reasons this is replaced by the equality of the fugacity/of each of the... 

Vapor-liquid phase equilibrium calculations have to be conducted for the estimation of solubility in the vapor phase (16,17). Alternatively, a cubic EOS can be applied for the estimation of properties of the liquid phase. The equality of fugacity in the two phases can be written as... 

At the three-phase equilibrium condition, the chemical potential (or fugacity) of water in the hydrate phase... 

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