Liquid, fugacity nonideal

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. 

Modified Raonlt s law includes the activity coefficientto accountfor liquid-phase nonidealities, but is limited by the assumption of vapor-phase ideality. This is overcome by introdnction of tire vapor-pliase fugacity coefficient. For species i in a vapor mixtnre, Eq. (11.48) is written ... 

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). 

For nonideal sterns, the fugacities of component i in the vapor and in the liquid play the same role as the component partial pressure In the vapor and the component vapor pressure in the liquid. The fugacity can be regarded as a thermodynamic pressure. For equilibrium, vapor fugacity is equal to liquid fugacity, i.e.,... 

Set out the relevant form of the thermodynamic definition of K value. At this low system pressure, the vapor-phase nonideality is uegUgible. Since neither component has a very high vapor pressure at the system temperature, aud siuce the differences between the vapor pressures and the system pressure are relatively small, the pure-liquid fugacities can be taken to be essentially the same as the vapor pressures. 

Chao-Seader Correlation. Reference was made earlier to the well known and much used Chao-Seader correlation for the prediction of vapor-liquid equilibrium for principally hydrogen-hydrocarbon systems with small amounts of CO2, H2S, O2, N2, etc. The heart of the correlation consists of several equations to represent liquid fugacity. The other two constituents, the Scatchard-Hildebrand equation for activity coefficients and the Redllch-Kwong equation for the vapor-phase nonideality, were already well established. 

If we choose to quantify the vapor-phase nonideality using the fugacity coefficient [Equation (7.5)] and the liquid-phase nonideality using the activity coefficient [Equation... 

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)...

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.

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. 

When t is zero, the relationship expresses essentially ideal behavior. Moving t to unity slowly introduces the nonideality expressed by liquid activity and the vapor fugacity coefficient. Taylor and others report solving some very difficult problems using this approach. 

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. 

It is interesting to note that the vapor and liquid compositions are usually different for ideal mixtures. We can see this from Eq. (6.6), since different pure component vapor pressures are rarely equal at the same temperature. This picture changes when nonideal mixtures are considered. As we see from Eq. (6.55, the vapor and liquid mole fractions can become equal when the fugacity and activity coefficients alter the pressure ratio enough to cause the K value to become unity. We then have an azeotrope. 

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. 

Be able to distinguish between ideal mixtures and nonideal mixtures (Scc. 9.3) Understand the concepts of excess properties and activity coefficients Be able to calculate the fugacity of a component in a vapor mixture and in a liquid mixture if an equation of state is available (Sec. 9.4)... 

As the first illustration of the use of these equations, consider vapor-liquid equilibrium in the hexane-triethylamine system at 60°C. These species form an essentially ideal mixture. The vapor pressure of hexane af this temperature is 0.7583 bar and that of triethylamine is 0.3843 bar these are so low that the fugacity coefficients at saturation and for the vapor phase can be neglected. Consequently, Eqs. 10.1-3 and 10.1-4 should be applicable to this system. The three solid lines in Fig. 10.1-1 represent the two species partial pressures and the total pressure, which were calculated using these equations and all are linear functions of the of liquid-phase mole fraction the points are the experimental results. The close agreement between the computations and the laboratory data indicates that the hexane-triethylamine mixture is ideal at these conditions. Note that this linear dependence of the partiaLand total pressures on mole fractions predicted by Eqs. 10.1-2 and 10.1-3 is trae only for ideal mixtures it is not true for nonideal mixtures, as we shall see in Sec. 10.2. 

Figure 11.3-3 shows the vapor-liquid and liquid-liquid equilibrium behavior computed for the system of methanol and n-hexane at various temperatures. Note that two liquid phases coexist in equilibrium to temperatures of about 43°C. Since liquids are relatively incompressible, the species liquid-phase fugacities are almost independent of pressure (see Illustrations 7.4-8 and 7.4-9), so that the liquid-liquid behavior is essentially independent of pressure, unless the pressure is very high, or low enough for the mixture to vaporize (this possibility will be considered shortly). The vapor-liquid equilibrium curves for this system at various pressures are also shown in the figure. Note that since the fugacity of a species in a vapor-phase mixture is directly proportional to pressure, the VLE curves are a function of pressure, even though the LLE curves are not. Also, since the methanol-hexane mixture is quite nonideal, and the pure component vapor pressures are similar in value, this system exhibits azeotropic behavior. 

In the analysis and correlation of vapor-liquid equilibrium (VLE) data it is essential, especially at superatmospheric pressures, to take into account the effect of vapor-phase nonideality. This is expressed by the fugacity coefficient which, as long as the density of the mixture is not greater than one fourth of its critical value, can be calculated reliably with the following equation (for a binary mixture) ... 

Figure 12.19 Effect of temperature on fugacity of a pure saturated liquid. Vapor-phase nonidealities (cpf) lower from the pure vapor-pressure curve, but the variation of /j-"with 1/T remains roughly linear. At supercritical temperatures, jnue vapor pressures do not exist nevertheless, for (0.9 < r /T < 1), we may choose the hypothetical pure liquid for the standard state and obtain a value of f° by extrapolation. These values were comjnited for pure water using data from steam tables [14].

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