Symmetrical ideal deviations from

The regular solution model (eq. 3.68) is symmetrical about xA = xB =0.5. In cases where the deviation from ideality is not symmetrical, the regular solution model is unable to reproduce the properties of the solutions and it is then necessary to introduce models with more than one free parameter. The most convenient polynomial expression with two parameters is termed the sub-regular solution model. 

A discrepancy in free enthalpy between the perfect solution and the non-ideal solution, if the reference system is symmetrical, is generally expressed by the excess free enthalpy GE, which consists of the enthalpy term HE and the entropy term -TSE i.e. GE = HE - TSE. Two situations arise accordingly in non-ideal solutions depending on which of the two terms, He and - TSE, is dominant The non-ideal solution is called regular, if its deviation from the perfect solution is caused mostly by the excess enthalpy (heat of mixing) HE ... 

In all cases, the structure can be described as edge-shared bioctahe-dral and none of the interbond angles deviate from idealized values by more than a few degrees. The Bi-Br distances to the bridging bromines, in the examples given, are longer than those to the terminal bromines, as expected, but in all cases, the Bi-Br-Bi bridging units are quite symmetrical. 

The same relationships hold for D /, symmetric molecules when ft, is set to zero (44). For both C3/, and D3), symmetry, the depolarization ratio is expected to be p = 1.5. However, the converse need not be true a depolarization ratio of close to 1.5 does not prove a point group with a threefold rotational axis. The molecule may be conformationally flexible then HRS detects a superposition of (ft) of the different conformers. A case in point is l,3,5-trinitro-2,4,6-triisopropylaminobenzene which exists as an interconverting mixture of strongly distorted boat and twist-boat forms (Wolff etal., 1993), with approximate C5 and C2 symmetry, respectively, but still shows a depolarization ratio of close to 1.5 (Verbiest et al., 1994 Wortmann et ai, 1997). Even in the case of conformational homogeneity it may still strongly deviate from the ideal symmetry provided the impact of conformational distortion on the electronic properties is not great. 

This chapter presents a brief review of some special aspects of the topological theory of molecular shape, with emphasis on various treatments of approximate symmetry. If a molecular arrangement has some nontrivial symmetry, then this symmetry can be exploited for the simplification of the study and prediction of many physical and chemical properties of the molecule. In many cases, however, the molecular arrangements may deviate from their ideal symmetry, yet many molecular properties remain similar to those present in the symmetric arrangement. Approximate symmetry, symmetry deficiency, the methods for their quantification, and various algebraic treatments of approximate symmetries preserving some of the features of the standard group theoretical description of point symmetry, are the subject of this chapter. 

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 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. 

We now turn to some simple examples where Aab can be calculated. We still stay within the first-order deviation from ideal symmetrical solution. 

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... 

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... 

For symmetrical frequency distributions of quantitative data, an ideal measure of dispersion would take into account each value s deviation from the mean and provide a measure of the average deviation from the mean. Two such statistics are the sample variance, which is the sum of squared deviations from the mean QZ(Y — Y)2) divided by n — 1 (where n is the number of data values), and the sample standard deviation, which is the positive square root of the sample variance. 

Within the range of concentrations for which the Fuoss-Onsager equation is expected to be valid, this equation accounts well for the effects of non-ideality in solutions of symmetrical electrolytes in which there is no ion association. It can thus be taken as a base-line for non-associated electrolytes and any deviations from this predicted behaviour can be taken as evidence of ion association (see Section 12.12). 

Often, our interest will lie not so much in the actual structure of a particular molecule or molecular fragment in a particular environment as in the details of how this molecule or fragment deviates from some reference structure with the same atomic connectedness or constitution. Insofar as we can usually ignore the absolute position and orientation of our molecule, comparisons of this kind are most conveniently made in terms of internal coordinates. The distortion can be expressed in terms of a total displacement vector D = pj, where the d/s are displacements along some set of basis vectors pj. The only difference to the internal coordinates described in the previous section is that for deformation coordinates the displacements dj are defined to be zero for the reference structure. This could be an observed structure, or a calculated one, or an idealized, more symmetric version of the structure we are interested in. 

The Porter equation is the simplest realistic expression for gE- It is appropriste for "symmetrical binary mixtures shewing small deviations from ideality, for example, the acetone-methanol system depicted in Fig. 1.4-1,... 

The function p7( ) is a symmetric bell-shaped curve centered at T, and pT( ) is narrower when the effective potential barrier is wider. For an ideal dynamical bottleneck kt is unity deviations from unity indicate that recrossing or other multidimensional effects are important. 

Because Pd and Au both have a face-centered cubic (fee) crystal lattice in the bulk, one should expect 0/,-symmetric clusters to be more stable than 7/,-symmetric species, at least for larger clusters. For clusters with up to 147 atoms, no clear preference was found and differences in average binding energies per atom amount to only some hundredths of an Deviations from idealized cluster geo-... 

C. Recent Fluid Mixture Theory.—The system CF4 + CH4 is at present unique among binary mixtures because (i) it is composed of simple non-polar spherically symmetric molecules, (ii) the system exhibits considerable deviations from ideality, and (iii) and x known explicitly and with high accuracy. G and F have been measured with high precision and (and hence TS ) can be estimated with somewhat lower precision. This binary system has therefore been used to assess the merits of various rival statistical theories of fluid mixtures. Some of these theoretical predictions are shown in Table 3. It is seen... 

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. 

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. 

Relatively large simulation boxes (>6 nm) may be necessary for systems with large molecules and/or in aggregating systems. The need for a large box may be anticipated if the experimental G, s show large deviations from their symmetric ideal values (Ploetz, Bentenitis, and Smith 2010b). 

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