Solutions ideal and nonideal

Many properties of solutions can be understood through thermodynamics. For example, we can understand how the boiling and freezing points of a solution change with composition, how the solubility of a compound changes with temperature and how the osmotic pressure depends on the concentration. 

We begin by obtaining the chemical potential of a solution. The general expression for the chemical potential of a substance is T) = p°(po5 T)+ 

RT In a in which a is the activity and p° is the chemical potential of the standard state in which a =. For an ideal gas mixture, we have seen (6.1.9) that the chemical potential of a component can be written in terms of its mole fraction Xk as Xk) = i1 p,T)+RT nxk. As we shall see in this section, 

In equation (8.1.1), if the mole fraction of the solvent jCs, is nearly equal to one, i.e. for dilute solutions, then for the chemical potential of the solvent the reference state T) may be taken to be p (j3, T), the chemical potential of the pure solvent. For the other components Xk 1 for these minor components, equation (8.1.1) is still valid in a small range, but in general the reference state is not T). Solutions for which equation (8.1.1) is valid over all values of Xk are called perfect solutions. When Xk = 1, since we must have PkiP T) = p (p, r), it follows that for perfect solutions 

The activity of nonideal solutions is expressed as Uk = Jk k in which y is the activity coefficient, a quantity introduced by G.N. Lewis. Thus the chemical 

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The behaviour of most metallurgically important solutions could be described by certain simple laws. These laws and several other pertinent aspects of solution behaviour are described in this section. The laws of Raoult, Henry and Sievert are presented first. Next, certain parameters such as activity, activity coefficient, chemical potential, and relative partial and integral molar free energies, which are essential for thermodynamic detailing of solution behaviour, are defined. This is followed by a discussion on the Gibbs-Duhem equation and ideal and nonideal solutions. The special case of nonideal solutions, termed as a regular solution, is then presented wherein the concept of excess thermodynamic functions has been used. 

Involving the Separation of Ideal and Nonideal Solutions by Equilibrium Stage Processes, Proc. 5th Symposium on Computers in Chemical Engineering, Vysoke Tatry, Czechoslovakia, Oct. 5-9 (1977). 

In Chaps. 2 through 5, the theta methods and variations of the Newton-Raphson method are applied to all types of single columns and systems of columns in the service of separating both ideal and nonideal solutions. Applications of the techniques presented in Chaps. 2 through 5 to systems of azeotropic and extractive distillation columns are presented in Chap. 6. An extension of these same techniques as required for the solution of problems involving energy exchange between recycle streams is presented in Chap. 7. Special types of separations wherein the distillation process is accompanied by chemical reactions are treated in Chap. 8. 

The difference between the Gibbs free energies of mixing of ideal and nonideal solutions is called the excess Gibbs free energy, which we shall denote by AGe. From (8.4.7) and (8.4.16) it follows that... 

Generating PCM in an amorphous state was neither able by varying the process conditions nor the type of solvent. This follows the proposed criterion of ideal and nonideal solutes and that a switch between precipitation and crystallization is not possible by just changing the process parameters. Therefore, producing PCM in an amorphous state was realized by generating solid dispersions of PVP and the API at different polymer to API ratios. 

A FIGURE 12.15 Behavior of Ideal and Nonideal Solutions (a) An ideal solution follows Raoult s law for both components, (b) A solution with particularly strong solute-solvent interactions displays negative deviations from Raoult s law. (c) A solution with particularly weak solute-solvent interactions displays positive deviations from Raoult s law. (The dashed lines in parts b and c represent ideal behavior.)... 

If the diffusion coefficients D are known, a plot of Z vs. 0)" can be developed to determine unknown bulk concentration C and the differential of potential per concentration for ideal and nonideal solutions [9, p. 59] is ... 

Explain the important distinctions between each pair of terms (a) molality and molarity (b) ideal and nonideal solution (c) unsaturated and supersaturated solution (d) fractional crystallization and fractional distillation (e) osmosis and reverse osmosis. 

There are ideal solutions, ideally dilute solutions, and nonideal solutions. Ideal solutions are solutions made from compounds that have similar properties. In other words, the compounds can be interchanged within the solution without changing the spatial arrangement of the molecules or the intermolecular attractions. Benzene in toluene is an example of a nearly ideal solution because both compounds have similar bonding properties and similar size. In an ideally dilute solution, the solute molecules are completely separated by solvent molecules so that they have no interaction with each other. Nonideal solutions violate both of these conditions. On the MCAT, you can assume that you are dealing with an ideally dilute solution unless otherwise indicated however, you should not automatically assume that an MCAT solution is ideal. 

When analyzing thermodynamic properties of polymer solutions it is sufficient to consider only one of the components, which for reasons of simplicity normally is the solvent. It is convenient to separate Eq. (3.88) for solvent (i = 1) into ideal and nonideal (or excess) contributions by defining... 

The derivation of Equations (435) and (436) from dilute solutions is only approximate. It is also possible to thermodynamically derive more fundamental types of these equations by using the activity concept, but initially we need to define non-localized, ideal and nonideal monolayers. If all the solute molecules are mobile in a monolayer, this is called a non-localized monolayer. We may consider three types of molecules in non-localized mono-layers ideal point molecules having no mass and volume where no lateral interactions are present between these point molecules non-ideal molecules having their mass and volume but no lateral interactions taking place between them, as above and non-ideal molecules having their mass and volume, and in addition appreciable lateral interactions taking place between them. 

The model predicts very high rejection (Fig. 4.8 B) for both the ideal and nonideal cases above about 10 bar, as observed experimentally. The mass transfer coefficient seems to have a negligible effect on the rejection, as observed for docosane. There is a discrepancy between nonideal and ideal model data for pressures under 10 bar. If activity coefficients are included, the model predicts -100% rejection for nearly aU pressures, only deviating sHghtly from 100% at very low pressures (-2 bar). If activity coefficients are not included, the rejection begins to deviate from 100% at around 8 bar and decreases to -60% as the pressure decreases to 4 bar, where the total flux becomes nearly zero. This behavior for the ideal solution case is due to the fact that the model predicts that the solvent flux drops considerably at pressures lower than 6 bar, while the solute flux does not change so dramatically, thus forcing the rejection to drop. 

The difference in the total free energies for the ideal and nonideal cases is termed the excess free energy of the nonideal solution Fj (31). Thus,... 

Figure 2.7. Osmotic compressibility (n/Hijeai) plotted as a function of for the ideal solution (dashed line) and nonideal solutions with N = 100 and x = 0.4, 0.5, and 0.55 (solid lines).

A solution which obeys Raoult s law over the full range of compositions is called an ideal solution (see Example 7.1). Equation (8.22) describes the relationship between activity and mole fraction for ideal solutions. In the case of nonideal solutions, the nonideality may be taken into account by introducing an activity coefficient as a factor of proportionality into Eq. (8.22). 

Since the infinite dilution values D°g and Dba. re generally unequal, even a thermodynamically ideal solution hke Ya = Ys = 1 will exhibit concentration dependence of the diffusivity. In addition, nonideal solutions require a thermodynamic correction factor to retain the true driving force for molecular diffusion, or the gradient of the chemical potential rather than the composition gradient. That correction factor is ... 

This approach to solution chemistry was largely developed by Hildebrand in his regular solution theory. A regular solution is one whose entropy of mixing is ideal and whose enthalpy of mixing is nonideal. Consider a binary solvent of components 1 and 2. Let i and 2 be numbers of moles of 1 and 2, 4>, and 4>2 their volume fractions in the mixture, and Vi, V2 their molar volumes. This treatment follows Shinoda. ... 

A hypothetical solution that obeys Raoult s law exactly at all concentrations is called an ideal solution. In an ideal solution, the interactions between solute and solvent molecules are the same as the interactions between solvent molecules in the pure state and between solute molecules in the pure state. Consequently, the solute molecules mingle freely with the solvent molecules. That is, in an ideal solution, the enthalpy of solution is zero. Solutes that form nearly ideal solutions are often similar in composition and structure to the solvent molecules. For instance, methylbenzene (toluene), C6H5CH, forms nearly ideal solutions with benzene, C6H6. Real solutions do not obey Raoult s law at all concentrations but the lower the solute concentration, the more closely they resemble ideal solutions. Raoult s law is another example of a limiting law (Section 4.4), which in this case becomes increasingly valid as the concentration of the solute approaches zero. A solution that does not obey Raoult s law at a particular solute concentration is called a nonideal solution. Real solutions are approximately ideal at solute concentrations below about 0.1 M for nonelectrolyte solutions and 0.01 M for electrolyte solutions. The greater departure from ideality in electrolyte solutions arises from the interactions between ions, which occur over a long distance and hence have a pronounced effect. Unless stated otherwise, we shall assume that all the solutions that we meet are ideal. 

Fig. 10 Dependence of vapor pressure of a solution containing a volatile solute, illustrated for (A) an ideal solution and (B) a nonideal solution and shown as a function of mole fraction of the solute. Individual vapor pressure curves are shown for the solvent (0) the solute ( ), and for the sum of these (X).

To overcome the problem of non-ideality the work be carried out at the Q temperature because in nonideal solutions the apparent Molecular weight is a linear function of concentration at temperatures near Q and the slope depending primarily on the second virial coefficient. 

The thermodynamic development above has been strictly limited to the case of ideal gases and mixtures of ideal gases. As pressure increases, corrections for vapor nonideality become increasingly important. They cannot be neglected at elevated pressures (particularly in the critical region). Similar corrections are necessary in the condensed phase for solutions which show marked departures from Raoult s or Henry s laws which are the common ideal reference solutions of choice. For nonideal solutions, in both gas and condensed phases, there is no longer any direct... 

An important attribute of Equation 5.16 is that the pressure exerted on both phases, Ptot, is common to both isotopomers. The important difference between Equations 5.16 and 5.9 is that the isotopic vapor pressure difference (P/ — P) does not enter the last two terms of Equation 5.16 as it does in Equation 5.9. Also isotope effects on the second virial coefficient AB/B = (B — B)/B and the condensed phase molar volume AV/V are significantly smaller than those on AP/P ln(P7P). Consequently the corrections in Equation 5.16 are considerably smaller than those in Equations 5.9 and 5.10, and can sooner be neglected. Thus to good approximation ln(a") is a direct measure of the logarithmic partition function ratio ln(Qv Q7QvQcO> provided the pressure is not too high, and assuming ideality for the condensed phase isotopomer solution. For nonideal solutions a modification to Equation 5.16 is necessary. 

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