Fugacity in the Vapor Phase

In the case of three-phase equilibria, it is also necessary to account for the solubility of hydrocarbon gases in water. This solubility is proportional to the partial pressure of the hydrocarbon or, more precisely, to its partial fugacity in the vapor phase. The relation which ties the solubility expressed in mole fraction to the fugacity is the following ... 

Estimate the flash-point of acetone and compare it with the experimental value given in the literature. Hint Start with the basic principle that the fugacity in the vapor phase must equal that in the liquid phase. The lower flammable limit for acetone is 2.55 percent by volume. 

Figure 2.8 (a) and (b) CO2 concentration in a ternary PMMA-C2H4-CO2 mixture as a function of CO2 partial pressure. The ethylene fugacity in the vapor phase was held constant during the measurements. The solid line represents the NELF prediction based on the binary mixtures data. Experimental data are from ReO" ... 

In Section 7.3, we explored the use of equations of state to calculate the fugacity in the vapor phase by means of the fugacity coefficient. This approach can be used in the liquid... 

In order to solve Equation (7.6) for the composition of species i in each phase at equilibrium, we developed expressions for the fugacity in the vapor phase and in the condensed phases. The difference in the expression for fugacity between vapor and condensed phases typically lies in the choice of reference state. 

The fugacity in the vapor phase is commonly calculated using an ideal gas reference state. Thus, the reference state for pure species i is at a pressure low enough that it behaves as an ideal gas and at the temperature of the system, as restricted by the definition in Equation (7.3). For species i in a mixture, we also specify that the reference state is at the composition of the mixture. We can formulate the fugacity in terms of the fugacity coefficient—a dimensionless quantity that compares the fugacity of species i to the partial pressure species i would have in the system as an ideal gas ... 

At pressures to a few bars, the vapor phase is at a relatively low density, i.e., on the average, the molecules interact with one another less strongly than do the molecules in the much denser liquid phase. It is therefore a common simplification to assume that all the nonideality in vapor-liquid systems exist in the liquid phase and that the vapor phase can be treated as an ideal gas. This leads to the simple result that the fugacity of component i is given by its partial pressure, i.e. the product of y, the mole fraction of i in the vapor, and P, the total pressure. A somewhat less restrictive simplification is the Lewis fugacity rule which sets the fugacity of i in the vapor mixture proportional to its mole fraction in the vapor phase the constant of proportionality is the fugacity of pure i vapor at the temperature and pressure of the mixture. These simplifications are attractive because they make the calculation of vapor-liquid equilibria much easier the K factors = i i ... 

The fugacity fT of a component i in the vapor phase is related to its mole fraction y in the vapor phase and the total pressure P by the fugacity coefficient ... 

As discussed in Chapter 3, at moderate pressures, vapor-phase nonideality is usually small in comparison to liquid-phase nonideality. However, when associating carboxylic acids are present, vapor-phase nonideality may dominate. These acids dimerize appreciably in the vapor phase even at low pressures fugacity coefficients are well removed from unity. To illustrate. Figures 8 and 9 show observed and calculated vapor-liquid equilibria for two systems containing an associating component. 

These were converted from vapor pressure P to fugacity using the vapor-phase corrections (for pure components), discussed in Chapter 3 then the Poynting correction was applied to adjust to zero pressure ... 

The fugacity coefficient departure from nonideaHty in the vapor phase can be evaluated from equations of state or, for approximate work, from fugacity/compressibiHty estimation charts. References 11, 14, and 27 provide valuable insights into this matter. 

The limits of the Lewis fugacity rule are not determined by pressure but by composition the Lewis rule becomes exact at any pressure in the limit as y( - 1, and therefore it always provides a good approximation for any component i which is present in excess. However, for a component with small mole fraction in the vapor phase, the Lewis rule can sometimes lead to very large errors (P5, R3, RIO). 

Furthermore, 5 vapor-liquid phase equilibria are involved, e.g. for molecular NH3, C02, H2S and S02 and for water. Applying the concept of Henry s constant Hi for the solution of a gas i in pure water and fugacity-coefficient for describing the influence of inter-molecular forces in the vapor phase, the resulting equations are ... 

The thermodynamic reaction equilibrium constant K, is only a function of temperature. In Equation 4.18, m, the activity of the guest in the vapor phase, is equal to the fugacity of the pure component divided by that at the standard state, normally 1 atm. The fugacity of the pure vapor is a function of temperature and pressure, and may be determined through the use of a fugacity coefficient. The method also assumes that an, the activity of the hydrate, is essentially constant at a given temperature regardless of the other phases present. 

When the liquid phase is ideal, Ki depends only on the temperature, the pressure, and the vapor composition. The procedure for determining the dew point in such a case is to (1) guess a temperature (2) calculate the Kt, which equal f°/iP, where is the fugacity of pure liquid i at the system temperature and pressure, , is the fugacity coefficient of the i th species in the vapor phase, and P is the system pressure and (3) check if the preceding dew-point equation is satisfied. If it is not, repeat the procedure with a different guess. 

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... 

Here fugacity coefficient of component i, P is the pressure, and S is the mole fraction of component i. Fugacity coefficients are usually used only for the vapor phase, so yj is usually meant to represent the mole fraction of component i in the vapor phase and Xj is usually reserved to represent the mole fraction in the liquid phase. Equation (2B-6) can be used with any equation of state to calculate the fugacity of the components in the mixture in any phase as long as the equation of state is accurate for the conditions and phases of interest. An equation of state that is explicit in pressure is required to use Equation (2B-6). 

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