Small Deviations from Dilute Ideal Solutions

SMALL DEVIATIONS FROM DILUTE IDEAL SOLUTIONS... 

In this chapter, we are mainly concerned with comparing small deviations from dilute ideal behavior in water and in nonaqueous solutions. A systematic study of this topic has never been undertaken, either because of lack of stimulating motivation or because of experimental difficulties. New interest in this topic has been aroused only recently with the realization that a central problem in biochemistry, the so-called hydrophobic interaction, can be intimately related to the problem of small deviations from dilute ideal solutions. This brought a new impetus to the study of this entire area. 

Small Deviations from Dilute Ideal Solutions... 

Chapter 6 is the extension of Chapter 5 to include mixtures of two or more liquids. The most important concepts here are ideal behavior and small deviations from it. Most of the treatment is based on the Kirkwood-Buff theory of solutions. The derivation and a sample application of this powerful theory are presented in detail. We also present the elements of the McMillan-Mayer theory, which is more limited in application. Its main result is the expansion of the osmotic pressure in power series in the solute density. The most useful part of this expansion is the first-order deviation from ideal dilute behavior, a result that may also be obtained from the Kirkwood-Buff theory. 

Few liquid mixtures are actually ideal over their entire composition range. Figure 33.2 illustrates two cases where the vapour pressure of liquid mixtures (solutions) deviate from Raoult s Law (Frame 32 and this frame, equations (33.3) and (33.4)) (positively A/B or negatively C/D) over the composition range but shows (see caption to figure) the end composition members (both representing cases of dilute solutions) do follow Raoult s Law for a limited, small, composition range. 

By employing the osmotic coefficient (which incidentally could also be called the boiling point or freezing point coefficient) we lose some of the formal resemblance to the equations of ideal solutions, but on the other hand is much more sensitive to deviations from ideality. Thus in dilute solutions is close to unity and under these conditions yi approaches unity (c/. 20.5, 20.6) and In y is almost zero. On the other hand division of In y by In a which is also very small may lead to quite large values of - 1. 

The first term on the right-hand side is the change of Gibbs free energy produced by dilution of the ions as ideal solutes. The second term takes care of the effect oil the Gibbs free energy of the deviations of the ions from ideal solutes. Let us take C% so small that the corresponding value of fs is unity, then... 

The Wilson equation is very effective for dilute composition where entropy effects dominate over enthalpy effects. The Orye-Prausnitz form of the Wilson equation for the activity coeflicient, as given in Table 5.3, follows from combining (5-2) with (5-41). Values of A ij < 1 correspond to positive deviations from Raoult s law, while values of A >1 correspond to negative deviations. Ideal solutions result from A, = 1. Studies indicate that Af, and ka are temperature dependent. Values of viJvjL depend on temperature also, but the variation may be small compared to temperature effects on the exponential term. 

Since the molecular weight of a polymer is usually at least three orders of magnitude greater than that of the solvent, for a small weight fraction of the solvent N, (otNj, consequently the mole fraction of the solvent approaches unity very rapidly. This, in effect, means that following Raoult s law, the partial pressure of the solvent in the solution should be virtually equal to that of the pure solvent over most of the composition range. Available experimental data do not confirm this expectation even if volume fraction is substituted for mole fraction. Polymer solutions exhibit large deviations from the ideal law except at extreme dilutions, where ideal behavior is approached as an asymptotic limit. 

When the concentration of the surfactant is gradually increased, one observes systematic deviations from the behavior of ideal dilute solution. This phenomenon may be ascribed to the formation of small aggregates, e.g., dimer, trimer, and so on, of solute molecules. This is a common phenomenon shared by many concentrated solutions. What makes aqueous surfactant solutions so remarkable is that over some small concentration range one finds an abrupt change in the properties of the solution. This phenomenon is ascribed to the formation of larger aggregates of solute molecules known as micelles. The... 

No mixture is perfectly ideal, and all real mixtures show deviations from Raoult s law. However, the deviations are small for the component of the mixture that is in large excess (the solvent) and become smaller as the concentration of solute decreases (Fig. 3.25). We can usually be confident that Raoult s law is reliable for the solvent when the solution is very dilute. More formally, Raoult s law is a limiting law (like the perfect gas law) and is strictly valid only at the limit of zero concentration of solute. 

No actual solutions are ideal, and many solutions deviate from ideal-dilute behavior as soon as the concentration of solute rises above a small value. In thermodynamics we try to preserve the form of equations developed for ideal systems so that it becomes easy to step between the two types of system. This is the thought behind the introduction of the activity, flj, of a substance, which is a kind of effective concentration. The activity is defined so that the expression... 

Determined in such way, the osmotic and activity coefficients of undissociated citric acid (m) and y(m) are presented in Table 2.14. As can be observed, the influence of temperature is rather small in the 30-45 °C range but it is much stronger at lower temperatures. In moderately concentrated citric acid solutions, values of (m) and y m) coefficients are nearly unity indicating that deviations from the ideal behaviour are minor. In very dilute solutions when all three steps of dissociation are involved, a quite different theoretical approach should be applied to evaluate osmotic and activity coefficients. Unfortunately, the lack of accurate and reliable experimental results in this concentration range prevents such calculations. 

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