Deviations from dilute ideal solutions

In the previous two sections we have discussed deviations from ideal-gas and symmetrical ideal solutions. We have discussed deviations occurring at fixed temperature and pressure. There has not been much discussion of these ideal cases in systems at constant volume or of constant chemical potential. The case of dilute solutions is different. Both constant, T, P and constant T, pB (osmotic system), and somewhat less constant, T, V have been used. It is also of theoretical interest to see how deviations from dilute ideal (DI) behavior depends on the thermodynamic variable we hold fixed. Therefore in this section, we shall discuss all of these three cases. 

We have already seen that the form of the expression for the chemical potential in the limit of dilute ideal solutions is the same in the three cases equations (5.56) and (5.57). 

Deviations from DI are observed whenever we increase the density of the solute pA, beyond the range for which (6.24) is valid. The extent of the deviations will, of course, depend on how, i.e., under which conditions, we add the solute. Keeping T, P or T, pB or T, V constant will result in different deviations. The general cases may be obtained by integrating equations (5.53)-(5.55) under the different conditions T, pB, or T, P, or T, V constant. For instance, in the open system (with respect to B) we have 

the excess chemical potential for this case is 

Similar but more complicated expressions may be obtained from (5.54) and (5.55). Clearly, since we do not know the dependence of Gap on pA we cannot perform these integrations. 

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For solutions (or rather mixtures) of s at higher concentrations beyond the realms of Henry s law, we depart from the traditional notion of solvation and must use the definition as presented in section 7.2. There exists a variety of data which measures the extent of deviation from dilute ideal solutions. These include tables of activity coefficients, osmotic coefficients, and excess functions. All of these may be used to compute solvation thermodynamic quantities. 

The formal theoretical framework for studying deviations from dilute ideal solutions is embodied in the so-called virial expansion of the osmotic pressure n), through the McMillan-Mayer (1945) theory of 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... 

The second source of data available for multicomponent mixtures are the excess thermodynamic quantities. These are equivalent to activity coefficients that measure deviations from symmetrical ideal solutions and should be distinguished carefully from activity coefficients which measure deviations from ideal dilute solutions (see chapter 6). In a symmetrical ideal (SI) solution, the... 

Fig. 2.4 The vapor pressure diagram of a dilute solution of the solute B in the solvent A. The region of ideal dilute solutions, where Raoult s and Henry s laws are obeyed by the solvent and solute, respectively, is indicated. Deviations from the ideal at higher concentrations of the solute are shown. (From Ref. 3.)...

Chemical Potential Representing Deviation from an Ideal Dilute Solution... 

If the attraction between the A and B molecules is stronger than that between like molecules, the tendency of the A molecules to escape from the mixture will decrease since it is influenced by the presence of the B molecules. The partial vapor pressure of the A molecules is expected to be lower than that of Raoult s law. Such nonideal behavior is known as negative deviation from the ideal law. Regardless of the positive or negative deviation from Raoult s law, one component of the binary mixture is known to be very dilute, thus the partial pressure of the other liquid (solvent) can be calculated from Raoult s law. Raoult s law can be applied to the constituent present in excess (solvent) while Henry s law (see Section 3.3) is useful for the component present in less quantity (solute). 

Osmotic pressure is often expressed by Equation 2.10, known as the Van t Hoff relation, but this is justified only in the limit of dilute ideal solutions. As we have already indicated, an ideal solution has ideal solutes dissolved in an ideal solvent. The first equality in Equation 2.9 assumes that yw is unity, so the subsequently derived expression (Eq. 2.10) strictly applies only when water acts as an ideal solvent (yw = 1.00). To emphasize that we are neglecting any factors that cause yw to deviate from 1 and thereby affect the measured osmotic pressure (such as the interaction between water and colloids that we will discuss later), riy instead of n has been used in Equation 2.10, and we will follow this convention throughout the book. The increase in osmotic pressure with solute concentration described by Equation 2.10 is... 

When deviations from the ideal zero-order theory of complete dissociation were noted in dilute solutions of strong electrolytes, attempts were made by Arrhenius and others to obtain agreement with experiment by assuming that the solution was ideal in all components, ionic as well as neutral, and that there was an equilibrium between undissociated electrolyte and dissociated ions. The theories based on these assumptions were of necessity unsuccessful. It has been experimentally determined that in very dilute solutions of strong electrolytes f 7 ln y+ is proportional to the square root of the sum over species of the product of the concentration of an ion and the square of its charge number. Thus, a representation in terms of associating species, which... 

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. 

Because of the simplicity of the functions of state of the ideal gas, they serve well as models for other mixing experiments. Dilute solutions, for example, can be modeled as ideal gases with the empty space between the gas atoms being filled with a second component, the solvent. In this case, the ideal condition can be maintained as long as the overall interaction between solvent and solute is negligible. Deviations from the ideal mixing are treated by evaluation of the partial molar quantities, as illustrated on the example of volume, V, in Fig. 2.25. The first row of equations gives the definitions of the partial molar volumes and Vg and shows the addition... 

In this case, in which we have continued our example of a regular solution having Wq = 2000Jmol", the solution does not deviate from the ideal very greatly until molalities well above 1 m. Activity coefficients based on mole Ifactions and molalities are shown in Table 8.3. The reason for two slightly different values of 7h> ynx nd y, is that molality is not exactly proportional to mole fraction except in the limit of infinite dilution, so that a Henryan tangent... 

The activity coefficient at infinite dilution accounts for deviations from the ideal form of Raoult s law. If the deviation is positive, y < 1 and the solute s retention time is shorter than that of an ideal solute. If the deviation is negative, then y > 1. and the retention time of the solute is increased. The adjusted retention volume and the pressure are related by the ideal-gas law ... 

We see that the hypothesis that electrolytes dissociate into ions, put forward by Arrhenius almost a hundred years ago, can account for the deviations from the ideal behavior of extremely dilute solutions. 

We are now interested in the first-order deviation from the ideal dilute limit. We assume that as we increase the salt concentration slightly beyond the ideal dilute limit, the long-range coulomb interaction between the ions causes the initial deviations from ideality i.e., at these concentrations the system would have been dilute ideal if the solute were not electrolytes. Thus, our main goal is to calculate the deviations due solely to the... 

We have started with the assumption that the micellar solution is an associated ideal dilute solution. When the aggregates are very large, deviations from this ideality cannot be ignored. Formally one can retain Eq. (8.9.11) but reinterpret 5Gn in (8.9.12) in terms of the solvation Gibbs energies of An and M not in pure water, but in a mixture of various aggregates. 

Walden s rule in electrochemistry describes the relationship between ionic conductivity o with viscosity r, that is, o x r = constant. 3 The electrochemical molar conductivity (Amp) can be calculated from ionic conductivity as Amp = oM/p, where M is the molecular weight and p is the density. The Walden plots of the molecular conductivity versus 1/in the unit of 0.1 Pas (=poise) proposed by Angell and coworkers24 are shown in Fig. 15 for the present four ILs. The Walden rule relates the molecular mobility (1/ tf) to the molar conductivity induced from the charged ions in solution electrolytes and characterizes ILs. The fully dissociated ions such as diluted aqueous KCl solution give a behavior shown as a line in Fig. 15. The deviation from the ideal plot, ZllVhas been proposed to relate with the ion pairing of ILs.25 In the present ILs, the ZlIV for EMIm-TFSA exhibited the smallest value, followed by DMPIm-TFSA, DEME-TFSA and Pis-TFSA. 

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