Nonideal miscibility

Nonideal miscibility - in ideal mixtures, the components do not interact with each other, thus the properties of the mixture depend linearly on the corresponding property of each component, and on the composition of the mixture. However, in practice, two-component lipid mixtures behave nonideally, and this gives rise to anomalies. For example the phosphatidylcholines dimyristoyl PC (DMPC, C14 saturated fatty acids), dipalmitoyl PC (DPPC, C16 saturated fatty acids) and distearoyl PC (DSPC, Cl 8 saturated fatty acids) have their main... 

Phase coexistence - based in part on the phenomenon of nonideal miscibility, but also for other reasons, real phase boundaries in the phase diagram are often not lines, but strips or areas in which two or more different phases coexist. Lipid transitions are not sharp, but have instead a finite width, i.e. they extend over a certain temperature range. Specific examples will also be presented below. 

Observations of thermotropic phase transitions even in simple lipid mixtures are complicated by factors such as nonideal miscibility, coexistence of different phases under certain conditions, and specific interactions between lipids, leading to complex formation. A well-studied, yet incompletely understood, binary lipid mixture that exemplifies these difficulties is the one formed by phosphatidylcholine and cholesterol. 

Similar to binary diffusivities, each element in the diffusivity matrix is expected to depend on composition, sometimes strongly, especially for highly nonideal systems. If the nonideality is strong enough to cause a miscibility gap, the eigenvalues would vary from positive to zero and to negative. If there is no miscibility gap, the eigenvalues are positive but can still vary with composition. 

Equation (3.6) illustrates that the solubility of a solid in a liquid depends on the enthalpy change at Tm and the melting temperature of the solid. Equation (3.6) is a valid one when T > Tm because the liquid solute in an ideal solution is completely miscible in all proportions. Table 3.1 shows the ideal solubilities of compounds and their heat of fusion. Equation (3.6) is the equation for ideal solubility. The relationship of In x2 (ideal or nonideal solubility) vs. 1/T is shown in Figure 3.1. 

The solubility of a liquid solute in octanol can be determined in terms of solubility parameters of the solute and octanol, as shown in Equation (3.20). However, the solubility parameters of liquid solutes (i.e., AHV) are generally unknown or are close to that of octanol and thus the liquid solutes are expected to be completely miscible with octanol. The critical solution temperature (Tc) of a nonideal solution of similar size solutes for complete miscibility is given by Hildebrand ... 

The compositions of the vapor and liquid phases in equilibrium for partially miscible systems are calculated in the same way as for miscible systems. In the regions where a single liquid is in equilibrium with its vapor, the general nature of Fig. 13.17 is not different in any essential way from that of Fig. I2.9< Since limited miscibility implies highly nonideal behavior, any general assumption of liquid-phase ideality is excluded. Even a combination of Henry s law, valid for a species at infinite dilution, and Raoult s law, valid for a species as it approaches purity, is not very useful, because each approximates real behavior only for a very small composition range. Thus GE is large, and its composition dependence is often not adequately represented by simple equations. However, the UNIFAC method (App. D) is suitable for estimation of activity coefficients. 

We conclude this discussion with one final reminder. The vapor-liquid equilibrium calculations we have shown in Section 6.4c are based on the ideal-solution assumption and the corresponding use of Raoult s law. Many commercially important systems involve nonideal solutions, or systems of immiscible or partially miscible liquids, for which Raoult s law is inapplicable and the Txy diagram looks nothing like the one shown for benzene and toluene. 

To illustrate the application of the film model for nonideal fluid mixtures we consider steady-state diffusion in the system glycerol(l)-water(2)-acetone(3). This system is partially miscible (see Krishna et al., 1985). Determine the fluxes Ap A2, and A3 in the glycerol-rich phase if the bulk liquid composition is... 

The NRTL equations— Eqnation (4.386), Eqnation (4.387), and Eqnation (4.391)—contain three parameters for each binary system that are adjusted to fit data. Experience indicates that a varies in the range 0.20 to 0.47. Where experimental data are scarce, the value of a can be set by referring to known valnes of similar mixtnres. Renon and Pransnitz [8] recommended values for broad classes of mixtures. A typical value of a is 0.3. The NRTL equation offers advantages over the Wilson eqnation for strongly nonideal mixtures, and especially for partially miscible systems. 

A so-called regular solution is obtained when the enthalpy change (AHmix) is nonideal (i.e., non-zero, either positive or negative) but the entropy change (A mix) is still ideal. So on the molecular level, while an ideal solution is one in which the different types of molecules (A and B, for example) behave exactly as if they are surrounded by molecules of their own kind (that is, all intermolecular interactions are equivalent), a regular solution can form only if the random distribution of molecules persists even in the presence of A-B interactions that differ from the purely A-A and B-B interactions of the original components A and B. This concept has proved to be very useful in the development of an understanding of miscibility criteria. 

In Fig. 9.26, the thermodynamic equilibrium, solid-liquid phase diagram of a binary (species A and B) system is shown for a nonideal solid solution (i.e., miscible liquid but immiscible solid phase). The melting temperatures of pure substances are shown with Tm A and Tm B. At the eutectic-point mole fraction, designated by the subscript e, both solid and liquid can coexist at equilibrium. In this diagram the liquidus and solidus lines are approximated as straight lines. A dendritic temperature T and the dendritic mass fractions of species (p)7(p)s and (p)equilibrium partition ratio kp is used to relate the solid- and liquid-phase mass fractions of species (p)7(p)J and (p)f/(p)f on the liquidus and solidus lines at a given temperature and pressure, that is,... 

Since its introduction in 1964, the Wilson equation, shown in binary form in Table 5.3 as (5-28), has received wide attention because of its ability to fit strongly nonideal, but miscible, systems. As shown in Fig. 5.5, the Wilson equation, with the binary interaction constants of A12 = 0.0952 and Aji = 0.2713 determined by Orye and Prausnitz, fits experimental data well even in dilute regions where variation of yi becomes exponential. Corresponding infinite-dilution activity coefficients computed from the Wilson equation are y" = 21.72 and 7 = 9.104. 

Equation (9-128) relates the difference in composition of the two liquid phases to the temperature T. Equation (9-128) is usually solved graphically by plotting w = tanh Z and w = lRTIa)Z and noting the points of intersection of the two curves. These curves are plotted in Fig. 9-13. If (IRTIoc) > 1, the two curves intersect only at the point m = 0 i.e., the two phases always have the same composition and, therefore, are miscible. Because of our symmetry assumption, X2 = Xi in this case. For (IRTIa) < 1, Eq. (9-128) also has a nonzero solution and the system is heterogeneous. Thus, a nonideal liquid solution will have a critical mixing temperature if a is positive. At the critical temperature, IX/2RT, = 1 and T, = o /2R. 

For our second nonideal system, we look at a process that has extremely nonideal VLB behavior and has a complex flowsheet. The components involved are ethanol, water, and benzene. Ethanol and water at atmospheric pressure form a minimum-boiling homogeneous azeotrope at 351K of composition 90mol% ethanol. Much more complexity is introduced by the benzene/water system, which forms two liquid phases with partial miscibility. The flowsheet contains two distillation columns and a decanter. There are two recycle streams, which create very difficult convergence problems as we will see. So this complex example is a challenging simulation case. 

If the top temperature is too cold and the bottom tenperature is too hot to allow sandwich conponents to exit at the rate they enter the column, they become trapped in the center of the column and accumulate there fKister. 20041. This accumulation can be quite large for trace conponents in the feed and can cause column flooding and development of a second liquid phase. The problem can be identified from the simulation if the engineer knows all the trace conponents that occur in the feed, accurate vapor-liquid equilibrium (VLE) correlations are available, and the simulator allows two liquid phases and one vapor phase. Unfortunately, the VLE may be very nonideal and trace conponents may not accumulate where we think they will. For example, when ethanol and water are distilled, there often are traces of heavier alcohols present. Alcohols with four or more carbons (butanol and heavier) are only partially miscible in water. They are easily stripped from a water phase (relative volatility 1), but when there is litde water present they are less volatile than ethanol. Thus, they collect somewhere in the middle of the column where they may form a second liquid phase in which the heavy alcohols have low volatility. The usual solution to this problem is to install a side withdrawal line, separate the intermediate component from the other components, and return the other components to the column. These heterogeneous systems are discussed in more detail in Chapter 8. 

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