Symmetric ideal solutions

There exist several reference states of solutions referred to as ideal state, for which we can say something on the behavior of the thermodynamic functions of the system. The most important ideal states are the ideal-gas mixtures, the symmetric ideal solutions and the dilute ideal solution. The first arises from either the total lack of interactions between the particles (the theoretical ideal gas), or because of a very low total number density (the practical ideal gas). The second arises when the two (or more) components are similar. We shall discuss various degrees of similarities in sections 5.2. The last arises when one component is very dilute in the system (the system can consist of one or more components). Clearly, these are quite different ideal states and caution must be exercised both in the usage of notation and in the interpretations of the various thermodynamic quantities. Failure to exercise caution is a major reason for confusion, something which has plagued the field of solution chemistry. 

Thus, the chemical potential, when expressed in terms of the intensive variables T, P and xA, has this explicit dependence on the mole fraction xA. A system for which relation of the form (5.27) is obeyed by each component, in the entire range of composition, is called a symmetrical ideal solution. It is symmetrical in the sense that from the assumptions (5.24) and (5.25), it follows that relation (5.27) holds true for any component in the system. In a two-component system, it is sufficient to define SI behavior for one component only. The same behavior of the second component, follows from the Gibbs-Duhem relation... 

The very fact that we make a distinction between the solute A and the solvent B means that the system is treated unsymmetrically with respect to A and B. This is in sharp contrast to the behavior of symmetrical 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. 

Symmetric ideal solution as a reference system. In the next case we assume that the two components A and B are similar in the sense of section 5.2, which means that... 

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... 

Thus, even when A and B are similar in the sense of (8.33), they can still have different affinities towards a third component. This was pointed out in both the original publication on the PS [Ben-Naim (1989, 1990b)] in two-component systems (see next section) as well as in Ben-Naim (1992). It was stressed there that similarity does not imply lack of PS. These are two different phenomena. Failing to understand that has led some authors to express their astonishment in finding out that symmetrical ideal solutions manifest preferential solvation. As we have seen above, SI behavior of the mixed solvents of A and B does not imply anything on the PS of s. This can have any value. In the next section, we shall see that the PS in two-component mixtures is related to the condition (8.33). However, the PS is not determined by the condition of SI solutions. In a three-component system, even when we assume the stronger condition of SI for the whole system, not only on the solvent mixture, i.e., when in addition to (8.33) we also have... 

Symmetrical ideal solution. Let A and B be similar particles, in the sense discussed in chapter 5. We perform the same process III as in figure 1.3. The corresponding change in Gibbs energy is (1.5). If we perform the same process, but under the same pressure before and after the mixing, we have for the chemical potentials of A and B, in the final states... 

We have seen in section 6.2 that the first-order deviations from symmetrical ideal solution have the form... 

Assuming that the two components form a symmetrical ideal solution, we can write ... 

Note that in contrast to the temperature derivative of the volume, here the second term on the right-hand side of (2.3.12) is independent of the sign of Hf — Hp. It does not matter in which direction the relative concentrations of the two components change all that matters is the size of the difference. Another important factor that appears both in (2.3.9) and (2.3.12) is the product XjXp. This particular product is a result of the assumption of symmetrical ideal solutions. Clearly, the same product would appear had we assumed that the system were dilute ideal, i.e. if either Xi 1 or Xp 1. In any case, it is clear that if either component is chosen in such a way that... 

We have discussed the specific model worked out by Wada, but the interpretation of the temperature dependence of the volume is the same for any other MM of water. Also, one should note that the assumption of symmetrical ideal solutions was explicitly introduced in this model. We shall further discuss the inappropriateness of this assumption in Sec. 2.3.4. 

We recall that condition (2.3.69) is essentially a condition on the similarity between the two species, in the sense that the local environments of L and H are similar. In a real mixture of two components A and B, similarity of the molecules (chemical composition structure, etc.) A and B implies similarity in their local environments as well. In the MM approach for water, we start with two components which are identical in their chemical composition. (Both L and H are water molecules ) Therefore, it is very tempting to assume that these two components are also similar in the sense of (2.3.69) hence, they form a symmetric ideal solution. However, since we require the two components to be very different in their local environments, they must be very dissimilar, thereby, invalidating the assumption of ideality. 

The general expression for the PMHC of s is quite involved and is omitted here [see Ben-Naim (1970b)]. Instead, we present here a simple example to demonstrate an important point. Suppose that the mixture of L and H forms a symmetrical ideal solution. Also, for simplicity, we assume that s is very dilute in water. The total heat capacity of the system can be written as (see Sec. 2.3)... 

As we have seen in Chapter 2, a large difference in the two components was also a requirement in the interpretation of the outstanding properties of pure water. Here, we should also bear in mind the relevance of this requirement to the assumption of ideality, often made in applying the MM to aqueous solutions. If the system is presumed to be a symmetrical ideal solution, then, in general, the species cannot be too different. On the other hand, if the assumption... 

Hence, in a symmetric ideal solution, the limiting PS, 5 and 5 3> have equal value, but are not necessarily zero. Thus, both A b and the components 5 and 8 3 are important in analyzing the sources of nonideality of the mixture. 

Equation 2.44 is the familiar form of the chemical potential of a symmetric ideal solution. We shall see in Section 2.6 that a mixture of hard rods always forms a symmetric ideal solution. 

Another issue that has been examined both numerically and theoretically is the deviation from symmetric ideal solution behavior, and its relation with the stability of the mixtures. It was shown that no miscibility gap can occur in such mixtures (Ben-Naim and Santos 2009). It should be noted that Equation 2.46 through Equation 2.48 in the original study contain some errors, although this did not affect the conclusions (Ben-Naim and Santos 2009). The second term on the right-hand side of Equation 2.46 should contain Xg instead of x in the numerator. This also affects the expressions provided in Equation 2.47 and Equation 2.49 of that paper. 

Relation (4.88) will be most useful for the study of symmetric ideal solutions carried out in the next section. 

SYMMETRIC IDEAL SOLUTIONS NECESSARY AND SUFFICIENT CONDITIONS... 

L and H are both water molecules. However, by their very definition, they differ markedly in local environment hence, it is unlikely that they obey the condition for symmetric ideal solutions. This point will be discussed further in Section 6.8, when we analyze a more general mixture-model approach to water. [See also the discussion following (6.52).]... 

This is the symmetrical ideal solution. We shall discuss the thermodynamics of ideal solutions further in Chapter 6. The average number of As (or Bs) in the system is obtained from... 

A simple case occurs when 2Wab aa + bb or q b = qAAQBB- (This corresponds to the symmetrical ideal solution of section 4.2.3 see also Chapter 6.) In this case,... 

Here we have expressed the chemical potential in terms of the intensive parameters T, P, and Xa. A system for which relation of the form (6.5.7) is obeyed by each component is called a symmetrical ideal solution. 

In the preceding section we derived the characteristic expression for the chemical potential in a symmetrical ideal solution. Here, we derive another ideal behavior, which is obtained whenever one component (the solute) is very dilute in the second component (the solvent). 

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