Dilute solutions small deviations from

Van t Hoff introduced the correction factor i for electrolyte solutions the measured quantity (e.g. the osmotic pressure, Jt) must be divided by this factor to obtain agreement with the theory of dilute solutions of nonelectrolytes (jt/i = RTc). For the dilute solutions of some electrolytes (now called strong), this factor approaches small integers. Thus, for a dilute sodium chloride solution with concentration c, an osmotic pressure of 2RTc was always measured, which could readily be explained by the fact that the solution, in fact, actually contains twice the number of species corresponding to concentration c calculated in the usual manner from the weighed amount of substance dissolved in the solution. Small deviations from integral numbers were attributed to experimental errors (they are now attributed to the effect of the activity coefficient). 

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. 

The last chapter deals with small deviations from very dilute solutions. The problem of hydrophobic interaction, considered to be of crucial importance in biochemical processes, is formulated, and methods of estimating the strength of solute-solute interaction in various solvents are discussed. Preliminary attempts at interpretation, based on concepts developed in the preceding chapters, are also surveyed. 

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. 

An important quantity whieh has been frequently studied is the mean ehain length, (L), and the variation of (L) with the energy J, following Eq. (12), has been neatly eonfirmed [58,65] for dense solutions (melts), whereas at small density the deviations from Eq. (12) are signifieant. This is demonstrated in Fig. 6, where the slopes and nieely eonfirm the expeeted behavior from Eq. (17) in the dilute and semi-dilute regimes. The predieted exponents 0.46 0.01 and 0.50 0.005 ean be reeovered with high preeision. Also, the variation of (L) at the threshold (p, denoted by L, shows a slope equal to... 

When solute and solvent are very dissimilar chemically, A is large. Therefore, deviations from infinitely-dilute-solution behavior are frequently observed for such mixtures at very small values of x2. For example, solutions of helium in nonpolar solvents show deviations from dilute solution behavior at values of x2 as low as 0.01. On the other hand, since both A and vf are usually positive, it sometimes happens that the last two terms in Eq. (65) tend to cancel each other, with the fortuitous result that Henry s law provides a good approximation to unexpectedly high pressures and concentrations (M3). 

In order to determine the ion pair dissociation constant Kd, of a salt it is necessary therefore to measure X as a function of C and obtain a roughly extrapolated value for X0. Calculation of the variables F(z)/X and f 2 F(z)CX is usually accomplished with a small computer program, and hence a more accurate value for X0 and a first value for Kd obtained from a straight line plot of these functions. It is, however, more convenient to carry out the whole process by computer with iteration accompanied by a least mean square calculation to obtained the most accurate value for X0 and Kd. For solvents of low dielectric constant, and if sufficiently dilute solutions are not examined, Fuoss plots deviate downward at higher concentrations, because of triple ion formation. This can lead to an excessively low estimate for X0 and too high a value for Kd. 

Considering the calculation of coil density, this is not surprising. A concentration of 1% in a solution seems small, but when the coil is more than 100 times diluted, considerable mutual penetrations of the coils can be expected, and, therefore, strong deviations from the behaviour of single, separate coils. 

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. 

If the buffered solution is dilute, this is its hydrogen-ion concentration. Because the activities of ions are affected by other ions, however, there is appreciable deviation from the calculated values in salt solutions as concentrated as 0,1 M, This fact accounts for the small discrepancies beuveen the pH values calculated from equilibrium con status and those gi en in the buffer tables. 

Eq. (6) is a rigorous equation for the solubility of poorly soluble solids in a mixed solvent. The only approximation involved is that the solubilities of the solid in either of the pure solvents and in the mixed solvent are very small (infinite dilution approximation). It is not applicable when at least one of these solubilities has an appreciable value. Indeed (see Table 1), when the solubility of a solute in a nonaqueous solvent exceeds about 5 mol%, such as the solubilities of drugs in propylene glycol (Rubino and Obeng, 1992), the deviation from the experimental data is about 11.5% for the Wilson equation, whereas the average deviation for all 32 mixtures of Table 1 is only 7.7%. 

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. 

Now we discuss the diameter (Mi + Mf) /2 of the coexistence curve. For a long time, the diameter anomaly in nonionic systems was a matter of controversy, because the deviations from rectilinear behavior are small, and there is an additional spurious 2(5 contribution, when an improper order parameter is chosen in data evaluation [104], The investigations of the picrate systems [72] and of the IL solutions [103] both yielded a substantial anomaly, consistent with an (1 — a) anomaly. Large diameter anomalies are expected, when the intermolecular interactions depend on the density [84], In the systems considered here, the dilute phase is essentially composed of ion pairs, while the concentrated phase is an ionic melt, which may explain the rather pronounced deviation from the rectangular diameter in the ionic systems. 

An example of a system exhibiting a small positive deviation from Raoult s law is a methanol-water solution (see fig. 1.5). It should be noted that when the concentration of methanol is small (xmcoh < 0.1), the vapor pressure of water is close to the value expected on the basis of Raoult s law. Similarly, for a dilute solution of water in methanol (xmcoh > 0.9, < 0.1), the vapor pressure of... 

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