Symmetrical ideal

The solute and the solvent are not distinguished normally in such ideal mixtures, which are sometimes called symmetric ideal mixtures. There are, however, situations where such a distinction between the solute and the solvent is reasonable, as when one component, say, B, is a gas, a liqnid, or a solid of limited solubility in the liquid component A, or if only mixtures very dilute in B are considered (xb 0.5). Such cases represent ideal dilute solutions. 

In considerations of the solid state, a natural starting point is high symmetry—a linear chain, a cubic or close-packed three-dimensional lattice. The orbitals of the highly symmetrical, idealized structures are easy to obtain, but they do not correspond to situations of maximum bonding. These are less symmetrical deformations of the simplest, archetype structure. 

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. 

We start by defining a symmetrical ideal (SI) solution as a system for which the chemical potential of each species has the form... 

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 symmetrical ideal behavior is equivalent to the well-known Raoult law. Suppose that a mixture of A and B is in equilibrium with an ideal-gas phase let PA be the partial pressure of A. The chemical potential of A in the gas phase is... 

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. 

The symmetric ideal (SI) solution is obtained for similar components in the sense that Aab = 0 for all compositions 0 < xA < 1, which again leads to (5.69)... 

The excess chemical potential, due to deviations from symmetrical ideal behavior, is thus... 

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. 

The second case, referred to as symmetric ideal (SI) solutions, occurs whenever the various components are similar to each other. There are no... 

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... 

It is interesting to note that if the mixed solvent of A and B forms a symmetrical ideal (SI) solutions, i.e., when... 

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... 

It is interesting to note that if the mixture is symmetrical ideal, then... 

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... 

In sections 6.6 and 6.7, we analyzed the conditions of stability using the parameter pAAB as a measure of the deviations from SI solutions. When Aab = 0> we had symmetrical ideal SI solutions. We found that for positive values of Aab, the system was always stable. For large negative values of A g, we found regions of instability. This conclusion seems to conflict with the experimental results that positive deviations from Raoult s law lead to instability. The classical examples shown in many books are mixtures of water and various normal alcohols. We reproduce the relevant curves in figure P. 1. Here, we plot the relative partial pressure of the alcohols in mixtures of water with methanol, ethanol, propanol and n-butanol. Clearly, in all of the four cases, deviations from Raoult s law as measured by either the quantities... 

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. 

A second possibility is to define L and H in such a way that the two components are similar, so that the solution is symmetrically ideal, ... 

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... 

Ideal solutions have proved to be a very useful reference point for the analysis of real data on solutions obtained by inversion of FST (Ben-Naim 2006). They can also be used to provide approximate behavior in situations where the required experimental data are not available. In principle, ideal behavior will depend on the concentration scale used. The most common type of ideal solution involves species that are fuUy miscible and where the composition is described by mole fractions. Ideal solutions of this type are characterized by chemical potentials of the form = d nx, together with zero excess enthalpies and volumes of solution for the whole composition range. This is also referred to as symmetric ideal (SI) behavior. Application of the above conditions in Equations 1.54 provides the following general relationships for any number of components (Ploetz, Bentenitis, and Smith 2010b) ... 

There are essentially three significant quantities that can be derived from the inversion of the KB theory. The first is a measure of the extent of deviation from symmetrical ideal (SI) solution behavior, A b. defined below in the next section. It also provides a necessary and sufficient condition for SI solution. The second is a measure of the extent of preferential solvation (PS) around each molecule. In a binary system of A and B, there are only two independent PS quantities these measure the preference of, say, molecule A to be solvated by either A or B molecules. Deviations from SI solution behavior can be expressed in terms of either the sum or difference of these PS quantities. Finally, the Kirkwood-Buff integrals (KBIs) may be obtained from the inversion of the KB theory. These provide information on the affinities between any two species for instance, PaGaa measures the excess of the average number of A particles around A relative to the average number of A particles in the same region chosen at a random location in the mixture. All these quantities can be obtained from the KB integrals. 

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. 

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