Symmetrical reference system

This symmetrical reference system gives us the activity coefficient that becomes unity as the molar fraction approaches unity for all constituent substances yl - 1 when xt - 1. 

The symmetrical reference system is based on Raoult s law in a perfect solution, while the unsymmetrical reference system is based on Henry s law in an ideal dilute solution. 

In this section we shall always define the activity coefficients with respect to the symmetrical reference system. Comparing Eq. 8.7 and Eq. 8.17, we define the excess free enthalpy (excess Gibbs energy) gE per mole of a non-ideal binary solution as Eq. 8.18 ... 

Instead of choosing an unsymmetrical reference system, we may employ a symmetrical reference system in the following manner. 

In this chapter we shall always define the activity coefficients with respect to the symmetrical reference system, (c/. chap. XXI, 3). 

This is an expression originally derived by John Ramshaw under certain approximations that one cannot expect to be exactly satisfied. However, as shown in Ref. 4, these approximations need not be assumed in the case of a cylindrically symmetric reference system to obtain (2.15). In contrast to (2.12), on the other hand, (2.15) will not hold if the reference system has more general symmetry. An Appendix to Ref. 5 gave a general reduction of (2.12) that yields an expression for e in terms of c(12) for all symmetries. Subsequently Ramshaw himself derived an equivalent general reduced form of (2.12) and (2.9) in somewhat different notation. 

By the same argument as before, the integral may be taken outside the summations for a symmetrical reference system (e.g. the RPM electrolyte) and applying the electroneutrality condition one sees that in this case Kj = K. Since the first two terms in the SL expansion form an upper bound for the free energy, the limiting law as T oo at constant c must always be approached from one side, unlike the Debye-Huckel limiting law which can be approached from above or below as c 0 at fixed temperature T (e.g. ZnSO and HCl in aqueous solutions). 

As mentioned in section 8.1, the value of the unitary chemical potential pi depends on the choice of the reference system. There are two reference systems which are commonly used one is unsymmetrical and the other is symmetrical. In discussing the reference systems we shall for convenience limit ourselves to a binary solution. 

This excess enthalpy hE corresponds to the heat of mixing of the non-ideal binary solution at constant pressure. Namely, hE = xf + x2h with - ht -h - -RT2(dlny JdT), where hf is the partial molar heat of mixing of substance i, ht is the partial molar enthalpy of i in the non-ideal binary solution, and h° is the molar enthalpy of pure substance i. Remind ourselves that the reference system for the activity coefficients is symmetrical. 

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 ... 

The symmetric parametrization can be achieved by taking as internal reference system the one that diagonalizes the inertia tensor, placing the principal axis in correspondence with that of maximal inertia [6,64], The symmetric hyperspherical coordinates can be calculated from the asymmetric hyperspherical coordinates ... 

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... 

As we mentioned in the opening paragraph, thermodynamic perturbation theory has been used in two contexts in applications to interaction site fluids. In this section, we will describe efforts to treat the thermodynamics and structure of interaction site fluids in terms of a perturbation expansion where the reference system is a fluid in which the intermolecular forces are spherically symmetric. In developing thermodynamic perturbation theories, it is generally necessary to choose both a reference system and a function for describing the path between the reference fluid and the fluid of interest. The latter choice is usually made between the pair potential and its Boltzmann factor. Thus one writes either... 

Linear transformations, including symmetry transformations in configuration space are analogous to those in Cartesian space (see Sections 1.2.2 and 1.2.3). For symmetrical reference structures, it is usually better to use not the internal coordinates themselves but to choose a new coordinate system in which the basis vectors are symmetry adapted linear combinations of the internal displacement coordinates with the special property that they transform according to the irreducible representations of the point group of the idealized, reference molecule (symmetry coordinates, see Chapter 2). 

For the simple potential given by (2.85), one can identify at this level of approximation the long-range parts of the correlations with their anisotropic parts and the short-range parts with their symmetric parts, and further identify the latter with pure reference system functions. Thus we have... 

The methods used are conveniently classified according to whether the reference system potential is spherically symmetric or anisotropic. Theories of the first type are most appropriate when the anisotropy in the full potential is weak and long ranged those of the second type have a greater physical appeal and a wider range of possible application, but they are more difficult to implement because the calculation of the reference-system properties poses greater problems. 

Roughly, the basic problem is the same as in the atomic case, but the practical difficulties are much more severe. A possible approach is to choose a reference-system potential Vo(r) spherically symmetric so that the integrations over the orientations (absent in atomic fluids) involve only the perturbation. The real system can be thus studied by a straightforward generalization of the A,-expansion developed for atomic fluids. 

Given the computations required for each individual chain, the number of scalar operations needed to compute the spatial acceleration of the reference membo, ao, is given in Table 6.3. Equation 6.38 is used to obtain the solution, which requires 0(m) spatial additions and a single 6x6 symmetric linear system solution. Thus, the number of opmtions required for ao is a function only of m, the number of chains in the simple closed-chain mechanism. The example of three chains (m s 3) is given in the last two columns of this table. 

As physical properties of the matter are independent of the chosen frame, suffixes a and p can be interchanged. Therefore, Xap = Xpot and only 6 components of x p are different, three diagonal and the other three off-diagonal. Such a symmetric tensor (or matrix) can always be diagonalized by a proper choice of the Cartesian frame whose axes would coincide with the symmetry axes of the LC phase. In that reference system only three diagonal components Xii, X22 and X33 are finite. 

For the symmetric blend the reference system is the homopolymer melt and hence the reference correlation functions are independent of species label (M, M ). It is important to emphasize that Eq. (8.11) is not valid for polymers of nonzero hard core thickness. Thus, the thread idealization is a very special limit characterized by a unique simplification of the integral equation theory. These equations have the analytical structure of a high temperature and/or... 

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