Moderately Dilute Ideal Solutions

In this section we discuss the properties of solutions which are not ideal throughout the entire range of composition but which are ideal only when moderately dilute. For these solutions, the chemical potentials of components 1,. . . , r can be written in the form 

Making use of Eqs. (6-54) and (7-142), we find that in the moderately dilute solution 

The quantity Hi(v°) is the partial molal enthalpy (molal volume) of component / in the hypothetical infinitely dilute state. Thus the heat of solution from the pure liquids at constant pressure is 

We next consider the properties of the fugacities of the vapor phase in equilibrium with the moderately dilute ideal solution. For vapor-liquid equilibrium, 

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An example of the use of this method is given in Figure 3.5.7 for the methyl acetate(l) and cydohexane(2) binary system (Pividal et al. 1992) at 313 K. The infinite dilution activity coefficient of each component in the other is available for this binary pair, the mixture is nearly symmetric and deviates only moderately from ideal solution behavior = 4.81/4.54). The solutions of eqns. (3.5.9 to 3.5.11)... 

The case of reactions in solution can be treated in a similar manner. Thus it can be shown that moderation is always observed if one adds (or removes) a small quantity of one of the reacting solutes of an ideal dilute solution. This follows since in a very dilute solution Ui is small in comparison with n so that vjn is negligible in comparison with Vijni since v,- is not zero. 

For low pressures (a few atmospheres and lower) we can apply the ideal gas model for gases and ideal mixture models for liquids. This formulation is very common in reactor technology. In some cases at higher pressures, the pressure effect on the gas phase is important. A suitable model for these systems is to use an EOS for the gas phase, and an ideal mixture model for liquids. However, in most situations at low pressures the liquid phase is more non-ideal than the gas phases. Then we will rather apply the ideal gas law for the gas phase, and excess properties for liquid mixtures. For polar mixtures at low to moderate pressures we may apply a suitable EOS for gas phases, and excess properties for liquid mixtures. All common models for excess properties are independent of pressure, and cannot be used at higher pressures. The pressure effect on the ideal (model part of the) mixture can be taken into account by the well known Poynting factor. At very high pressures we may apply proper EOS formulations for both gas and liquid mixtures, as the EOS formulations in principle are valid for all pressures. For non-volatile electrol3d es, we have to apply a suitable EOS for gas phases and excess properties for liquid mixtures. For such liquid systems a separate term is often added in the basic model to account for the effects of ions. For very dilute solutions the Debye-Htickel law may hold. For many electrolyte systems we can apply the ideal gas law for the gas phase, as the accuracy reflected by the liquid phase models is low. 

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. 

For dissolved, neutral substances, it is normally an acceptable approximation to assume ideality in concentrations less than 0.1—1.0 mol/ . For ions in solutions, however, the assumption only applies to highly diluted solutions even at concentrations of 0.001—0.01 mol/ , ionic solutions can deviate noticeably from ideality. In the following, we shall briefly explain how it is possible in moderate concentrations of dissolved and dissociated salts to correct for this deviation from ideality. 

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