Nonideal gases fugacity

Equation 5.21 shows that the Langmuir constant is a direct function of the particle partition function within the cavity qj/, in particular Cjj contains the nonideal gas translation term. When the fluid in equilibrium with the hydrate is a nonideal gas, the pressure of component J in Equation 5.22a is replaced with its fugacity,//. 

Nonideal gas behavior is accounted for by introducing the fugacity coefficient of component 1 in the gas mixture. 

In principle this accomplishes the task one measures the (molar) volume of the nonideal gas as a function of the applied pressure at fixed temperature T, beginning at a very low, fixed value of the pressure, Pi, and ending at the pressure P for which the fugacity / is to be found. Alternatively, an equation of state may be employed for insertion in Eq. (3.1.10). If this is not convenient one may integrate by parts to obtain... 

These two effects are summarized in Table 20.4-8. Moreover, the percentage reduction in flux for each component dne to nonideal gas-phase effects is listed in Table 20.4-8. Ii is clear ther the redaction in fugacity driving force is very smell (0.5%) for methane, which tends to he an ideal gas under these conditions. For CO2, however, rhe effects amonnt 10 roughtly 7,5% and account for a roughly 6,9% reduction in the ides] separation factor. To reemphasize, these effects would be observed even if the polymer-phase sorption and transport behavior did not show dual-mode effects and were perfectly ideal. 

Nonideal solution effects can be incorporated into /f-value formulations in two different ways. Chapter 4 described the use of the fugacity coefficient, in conjunction with an equation of state and adequate mixing rules. This is the method most frequently used for handling nonidealities in the vapor phase. However, tv reflects the combined effects of a nonideal gas and a nonideal gas solution. At low pressures, both effects are negligible. At moderate pressures, a vapor solution may still be ideal even though the gas mixture does not follow the ideal gas law. Nonidealities in the liquid phase, however, can be severe even at low pressures. In Section 4.5, il was used to express liquid-phase nonidealities for nonpolar species. When polar species are present, mixing rules can be modified to include binary interaction parameters as in (4-113). 

Equilibrium Constant for Nonideal Gases. When the Equilibrium constant is applied to nonideal gas mixtures, the fugacities are used instead of partial pressures. Thus, the equilibrium equation for the reaction... 

For nonideal gas mixtures describable by a fugacity function (ex equation of state or the principle of corresponding states) ... 

Correct for vapor-phase nonidealities. To account for the effects of nonideal-gas behavior, we compute fugacity coefficients using the simple virial equation... 

From the stoichiometric point of view, /j.i is the chemical potential of a nonideal gas. Therefore, it can be assigned with a fugacity coefficient... 

These fugacity effects simply correspond to gas-phase driving force reductions resulting from nonideal gas-phase behavior. Such effects clearly will bMome more substantial as higher total system pressures are considered in CO -containing systems. Conversely, such effects will be relatively unimportant for mixtures containing only supercritical, relatively ideal components. 

Fortunately, these phenomena can be predicted readily by using standard thermodynamic equations of state to calculate the fugacity of each component in the upstream and downstream gas phases. For plasticization-prone polymers such as cellulose acetate, the depression of COj fugacity in high CH pressure situations may suppress large upswings in permeability noticed for pure CO2 (as in Fig. 20.3-9). The area of nonideal gas-phase effects clearly requires considerably more careful investigation. 

VAPOR-LIQUID EQUILIBRIUM RATIOS FOR IDEAL-SOLUTION BEHAVIOR 3.1 FUGACITY OF PURE LIQUID 3.3 NONIDEAL GAS-PHASE MIXTURES 3.4... 

In this expression for the chemical potential, the first addendum, U(,(T), is a standard potential at a fixed pressure. The second addendum expresses the contribution from the fugacity of the pure component. The third addendum is due to mixing. The dependence/ (< >, T) may be found from the Gibbs adsorption equation (13b), where the integration is often carried out from zero pressure (and, correspondingly, the value of 0 is equal to zero). With such an expression for the chemical potentials of the adsorbed phase, equilibrium conditions (12), for the equilibrium with a nonideal gas phase, are reduced to the form... 

Equation (2.4.15) relates the chemical potential of an ideal gas to / Tln P,- in accordance with (i), this suggests that for a real gas should be specified by an analogous expression, namely RT In i, where f is termed the fugacity of the ith constituent of the gas. To satisfy (ii), this quantity must converge on the pressure P, at ideality. Since /x,- is specified only to within an arbitrary constant, only the difference in chemical potential of the nonideal gas in two states, 1 and 2, may be uniquely specified as... 

If, as shown above, for ideal gas mixtures the fugacity of one species in the mixture is equal to its partial pressure, then we would like to extend that simple idea to nonideal gas mixtures, and to solutions of liquids and solids. We can, using the definition of an ideal solution. An ideal solution is like an ideal gas in the following respects ... 

In summary, the reference state for species i in the liquid (or solid) phase is no more than a particular state, real or hypothetical, at a given P and Xi (usually that of the system) and at the temperature of the system. We choose the reference state to be that of an ideal solution in which the fugacity is linearly proportional to mole fraction. While the concept of an ideal solution was conjured up in analogy to an ideal gas, there are some interesting differences. A pure gas can be a nonideal gas, while a pure liquid cannot be a nonideal solution because all intermolecular forces in a pure liquid are the same Additionally, an increase in pressure leads to deviations from ideal gas behavior, whereas deviations from ideal solution are caused by changes in composition because nonideal behavior results primarily from the chemical differences of species in a mixture, even at low pressures. 

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 ... 

Fig. 6.2 Conceptualization of the fugacity of a compound in a nonideal liquid mixture when gas and hquid phases are in equilibrium (Schwarzenbach et al. 2003)...

In contrast to the pressure of an ideal gas, the fugacity is not only a function of the amount of substance and temperature, but also of the composition (types and amounts of gaseous compounds present) of the gaseous system and of die total pressure. The fugacitiy of a gaseous compound is, however, closely related to its partial pressure. To account for the nonideality of the gas, one can relate these terms by using a fugacity coefficient, 0ig ... 

Figure 3.9 Conceptualization of the fugacity of a compound i (a) in an it/ea/ gas (h) in a pure liquid compound i (c) in an Wen/ liquid mixture and (d) in a nonideal liquid mixture (e.g., in aqueous solution). Note that in (b), (c), and (d), the gas and liquid phases are in equilibrium with one another.

In this definition, the activity coefficient takes account of nonideal liquid-phase behavior for an ideal liquid solution, the coefficient for each species equals 1. Similarly, the fugacity coefficient represents deviation of the vapor phase from ideal gas behavior and is equal to 1 for each species when the gas obeys the ideal gas law. Finally, the fugacity takes the place of vapor pressure when the pure vapor fails to show ideal gas behavior, either because of high pressure or as a result of vapor-phase association or dissociation. Methods for calculating all three of these follow. 

Related Calculations. If the gas is not ideal, the fugacity coefficients , will not be unity, so the activities cannot be represented by the mole fractions. If the pressure is sufficient for a nonideal solution to exist in the gas phase, , will be a function of y, the solution to the problem. In this case, the y, value obtained for the solution with Lewis-Randall rule for... 

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