Behavior of Ideal Mixtures

According to Dalton s law ofpartial pressures the partial pressirre of a component is proportional to its mole fraction in the gas and to the total pressure p  

Ideal behavior of gaseous mixtures requires low to moderate pressures ip 10 bar). A combination of these equations provides a correlation between mole fraction j in the gaseous phase and the total pressure p  

A combination of both equations leads to the relation for the thermodynamic equilibrium of a two-component mixture  

This equation is applicable for ideal liquids and vapor phases only. A liquid is ideal if Raoult s law can be applied for both components over the whole concentration range. 

The equation above can be plotted as the selectivity or equilibrium diagram given in Fig. 2.1-16. This diagram shows that the gaseous phase is always richer in light ends component than the liquid phase. The ratio of partial pressures of both components is called relative volatility a 

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However, in the study of thermodynamics and transport phenomena, the behavior of ideal gases and gas mixtures has historically provided a norm against which their more unruly brethren could be measured, and a signpost to the systematic treatment of departures from ideality. In view of the complexity of transport phenomena in multicomponent mixtures a thorough understanding of the behavior of ideal mixtures is certainly a prerequisite for any progress in understanding non-ideal systems. 

The freezing behaviors of ideal mixtures over the entire range of temperatures can be calculated readily. The method is explained for example by Walas (Phase Equilibria in Chemical Engineering, Butterworths, Stoneham, MA, 1985, Example 8.9). 

So little systematic information is available about transport in liquids, or strongly non-ideal gaseous mixtures, that attention will be limited throughout to the behavior of ideal gas mixtures. It is not intende thereby, to minimize the importance of non-ideal behavior in practice. 

Two relations are postulated to describe the P-V-T behavior of ideal gas mixtures ... 

The total pressure in the container is the sum of the partial pressures Aotal He + PO2 We have used He and O2 to illustrate the behavior of a mixture of ideal gases, but the same result is obtained regardless of the number and identity of the gases. 

Phase behavior of lipid mixtures is a much more difficult problem, due to nonideal mixing of lipid components. Ideal mixing implies like and unlike lipids have the same intermolecular interactions, while nonideal mixing results from differential interactions between lipid types. If the difference is too great, the two components will phase separate. While phase separation and lateral domain formation have been observed in many experiments, we lack a molecular-level physical description of the interactions between specific lipids that cause the macroscopic behavior. The chemical potential of a lipid determines phase separation, as phase coexistence implies the chemical potential of each type of lipid is equal in all phases of the system [3,4],... 

Figure 6. Non-ideal gas behavior of a mixture of hydrogen and helium (T 120K 90 and 10 molecules, respectively) and a 50 molecule sample of ammonia (T 298 K), both simulated with Odyssey.

As in the case of ideal gases, ideal liquid solutions do not exist. Actually, the only solutions which approach ideal solution behavior are gas mixtures at low pressures. Liquid mixtures of components of the same homologous series approach ideal-solution behavior only at low pressures. However, studies of the phase behavior of ideal solutions help us understand the behavior of real solutions. 

The behavior of a mixture is determined by a system of ordinary differential equations, while the required state, either equilibrium or stationary, is determined by a time-independent system of algebraic equations. Therefore, at first glance one would not expect any qualitative difference between the equilibrium and stationary states. Ya.B. shows that in the equilibrium case, even for an ideal system, a variational principle exists which guarantees uniqueness. Such a principle cannot be formulated for the case of an open system with influx of matter and/or energy. 

An excess property is the difference between the actual property value of a solution and the ideal solution value at the same composition, temperature, and pressure. Therefore, excess properties represent the nonideal behavior of liquid mixtures. The major thermodynamic properties for ideal mixtures are... 

The solid lines on Figure 4 take into account the nonideal behavior of adsorbed mixtures of ethylene and ethane in NaX. This system is highly nonideal because of the interaction of the quadrupole moment of ethylene with the soditun cations of NaX. Activity coefficients at infinite dilution are unity at the limit of zero pressure and 0.27 at high pressure. The dashed lines on Figure 4 were calculated for am ideal adsorbed solution (IAS) and the resulting error in the individual isotherm for ethane at 30 bar is 20%. 

Although the general phenomena and the qualitative results described in this section remain valid for any isotherm model, provided they are convex upward and do not intersect, the quantitative results of the shock layer theory presented in Chaptersl4 and 16 are valid only when the adsorption behavior of the mixture components is properly described by the competitive Langmuir isotherm model. The theory shows conclusively that, when the separation factor decreases, the shock layer thickness, hence the width of the mixed zone in the isotachic train, increases in proportion to oc + l)/ a — 1) (Eqs. 16.27a and 16.27b). At the same time, the column length required to reach isotachic conditions increases also indefinitely, as predicted by the ideal model. 

Introducing the molar fractions into Eq. (8.19), we obtain (we again assume the ideal behavior of the mixture)... 

Extending the same procedure for mixtures, say of two components, A and B will give us the second virial coefficient for a mixture. The first-order correction to the ideal-gas behavior of the mixture is... 

Neglecting the activity coefficient and hence assuming ideal behavior of the mixture leads to Raoult s law. 

Hauser J, Reinhardt GA, Stumm F, Heintz A (1988), Sorption Non-Ideal Behavior of Liquid Mixtures in PVA and its Influence on the Pervaporation Process. Third International Conference on Pervaporation Processes in the Chemical Industry. R Bakish (ed.). Bakish Material Corporation, Englewood, N), USA, 134. 

From equation (41) it is evident, of course, that the conditions may be very much more complicated if, for example, we do not work at constant pressure, or if, as may be the case in real gases, the internal energy of the system changes during the mixing. The particularly simple conditions chosen are, however, practically fulfilled in many cases by real gases they correspond to the behavior of ideal gases or ideal gas mixtures. 

We learn that the behavior of a mixture of gases can be understood by Dalton s law of partial pressures, which is an extension of the ideal gas equation. (5.6)... 

Minka and Myers (8) have extended the concept of surface excess and selectivity to multicomponent mixtures. They applied a theory of an ideal adsorbed phase to predict the adsorption behavior of ternary mixtures from adsorption measurements in binary systems. Having binary data in the form of Equation (10) a ternary isotherm is calculated as follows ... 

The ideal behavior of isotopic mixtures is expected if one assumes that the intermolecular forces between pairs of like molecules of each type and between unlike molecules are all the same, and further assumes the isotopic molecules to have the same size. Both assumptions are reasonable to a first approximation. Highly precise vapor pressure measurements on isotopic mixtures have shown, however, that even these mixtures exhibit small, but still significant, deviations from ideal behavior (Jancso et al. 1993, 1994). Theoretical analysis has demonstrated that the origin of nonideality is closely connected with the difference between the molar volumes of isotopic molecules ( molar volume isotope effect ). 

The quantitative relationship between vapor pressure and composition of homogeneous liquid mixtures is known as Raoult s law (Eq. 4.2). The factor represents the partial pressure of component X, and it is equal to the vapor pressure, P, of pure X at a given temperature times the mole fraction of X in the mixture. The mole fraction of X is defined as the fraction of all molecules present in the liquid mixture that are molecules of X. It is obtained by dividing the number of moles of X in a mixture by the sum of the number of moles of all components (Eq. 4.3). Raoult s law is strictly applicable only to ideal solutions, which are defined as those in which the interactions between like molecules are the same as those between unlike molecules. Fortunately, many organic solutions approximate the behavior of ideal solutions, so the following mathematical treatment applies to them as well. 

Since the activity coefficient of a species relates its actual behavior to its ideal behavior at the same T and p, let us begin by examining behavior in ideal mixtures. 

Finally it should be noted that the relationship between and is independent of the temperature. Thus all the non-ideal behavior of the mixture resides in the entropy of mixing, the heat of mixing being zero. 

A great deal of work with different experimental and theoretical techniques has been performed in the past on characterizing liquid binary mixtures of simple alcohols with water [46,58-78]. Water mixtures with a short-chain alcohol up to 1-propanol [50,58-81], which mix with water over the whole concentration regime, have been studied the most extensively. Nevertheless, one can also find some similar studies with long-chain alcohols that are practically insoluble in water [46,76,77,82-86]. The motivation for such abundant work lies in the anomalous behavior of such mixtures. Namely, when simple alcohols are mixed with water the entropy of the system seems to increase far less than would be expected for an ideal solution of randomly mixed molecules [75]— a phenomenon that is also clearly expressed in the anomalous behavior of some other measurable properties. Furthermore, one can find two completely different concepts to explain these anomalous properties existing in the literature, which makes these binary mixtures even more intriguing nowadays. 

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