Solutions, regular defined

These are related to the regular solutions (defined by their behavior as Riccati-Bessel functions/<, i r) jt kr) when r 0) by... 

The physical signiflcance of this result may be clearer if we return to the case of a regular solution defined by Eq. 6.3-6. Using the chemical potential derivative in Eq. 6.3-7, we find from Eq. 6.3-19 that... 

The solubility parameter 5 of a pure solvent defined initially by Hildebrand and Scott based on a thermodynamic model of regular solution theory is given by Equation 4.4 [13] ... 

The behaviour of most metallurgically important solutions could be described by certain simple laws. These laws and several other pertinent aspects of solution behaviour are described in this section. The laws of Raoult, Henry and Sievert are presented first. Next, certain parameters such as activity, activity coefficient, chemical potential, and relative partial and integral molar free energies, which are essential for thermodynamic detailing of solution behaviour, are defined. This is followed by a discussion on the Gibbs-Duhem equation and ideal and nonideal solutions. The special case of nonideal solutions, termed as a regular solution, is then presented wherein the concept of excess thermodynamic functions has been used. 

Both the binodal line, defining the immiscibility gap, and the spinodal line are for a regular solution symmetrical about xA = xB = 0.5. This is shown in Figure 5.7(a), where theoretical predictions of the miscibility gaps in selected semiconductor systems are given [15],... 

One of the approaches to calculating the solubility of compounds was developed by Hildebrand. In his approach, a regular solution involves no entropy change when a small amount of one of its components is transferred to it from an ideal solution of the same composition when the total volume remains the same. In other words, a regular solution can have a non-ideal enthalpy of formation but must have an ideal entropy of formation. In this theory, a quantity called the Hildebrand parameter is defined as ... 

An eigenfunction ijr of the Schrodinger equation satisfies the homogeneous integral equation b = f 3 Govt/r. In cell r this defines a locally regular solution of the Helmholtz equation... 

According to the regular solution theory, polymers are as a rule soluble in solvents if their solubility parameters are similar to those of the corresponding solvent. The upper limit for good solubility is defined to he a difference (AS) of 6 units between solubility parameters ... 

In this equation, Qm is the molar surface area, m i is a structural parameter defined in Section 1.1 (see Figure 1.3) and A is the regular solution parameter of Ni-Si alloy defined by equation (4.3). From the slope of the osL(XNi) curve for XNi— 0, the adsorption energy is found to be E i,(f ) = —8.2 kJ/mole. Thus, in equations (1.2), all the quantities are known (or can be easily estimated), except W and Wf 1 which represent respectively the work of adhesion and the work of immersion of pure liquid Ni in metastable equilibrium with SiC (i.e., for a supposed non-reactive pure Ni/SiC system). The values deduced from equation (1.2) are Wj4 = 3.17 J/m2 and W = —1.35 J/m2 for pure Ni. They are reported in Figure 7.6 along with the corresponding value of contact angle. 

A thermodynamic method, more fitting to this chapter, has been proposed by Nauman et al. They claim a process for the separation of a physically mixed solid polymers by selective dissolution. They rely on the different polymer solubility characteristics. Tables of this property have been reported and are based on regular solution theory and Hildebrand solubility parameters. The core of the Nauman invention is to find suitable solvents to dissolve particular polymers under defined temperature and pressure conditions. A mixture of polymers is first added to one solvent, at a given temperature, in order to dissolve a particular polymer. The remaining polymer mixture is then treated at a higher temperature with the same solvent or with a different solvent. For clarity, two examples are taken from the patent."... 

Regular solution theory characterises nonpolar solvents in terms of solubility parameter, 6y which is defined as... 

Regular solutions are mixtures of low molar mass species with Aa = Ab= 1. Polymer solutions are mixtures of macromolecules (Aa = A 1) with the low molar mass solvent defining the lattice (Ab = 1). Polymer blends are mixtures of macromolecules of different chemical species (Aa 1 and Ab > 1). 

Equation (6.27) is frequently used in the calculation of surface adsorption in dilute solutions. For strictly regular solutions, the activity coefficient is defined as... 

For many solutions, it is not possible to vary the composition of the components over the whole range of mole fractions. This is obviously true of solutions made up of a solid and a liquid. For these systems it is better to choose a standard state which is based on the properties of a dilute solution. This leads to the definition of an ideally dilute solution. Such a system is easily defined on a molecular basis as one in which the solute molecule only comes in contact with solvent molecules, and never with another solute molecule. In the previous discussion of regular solutions it was concluded that, when the two components are of equal size, the coordination number for the other molecules around a central one is twelve. This suggests that an ideally dilute solution must have a solute mole fraction which is less than 1/13, that is, 0.08. 

A simple thermodynamical model derived from the theory of regular solutions [26], can account for the main features of both continuous and discontinuous transitions. In this theoretical framework, so-called Slitcher-Drickamer model, the interaction term in equation (2), r( HsX defined as ) hs(J — hs) where y is an interaction parameter which reflects the tendency for molecules of one type to be surrounded by like molecules (y > 0). So, equation (5) becomes ... 

Traditionally, solubility parameters are given in (cal/cm ) = Hild(ebrands), in honor of the founder of the regular solution theory, Joel Hildebrand. Now, they are more commonly listed in (MPa) (= 1 (J/cm ) ). The solubility parameters were originally defined in conjunction with the regular solution theory, as discussed in Section 16.3.3.1. 

Most of the recent theories of liquid solution behavior have been based on well-defined thermodynamic or statistical mechanical assumptions, so that the parameters that appear can be related to the molecular properties of the species in the mixture, and the resulting models have some predictive ability. Although a detailed study of the more fundamental approaches to liquid solution theory is beyond the scope of this book, we consider two examples here the theory of van Laar, which leads to regular solution theory and the UNIFAC group contribution model, which is based on the UNIQUAC model introduced in the previous section. Both regular solution theory and the UNIFAC model are useful for estimating solution behavior in the absence of experimental data. However, neither one is considered sufficiently accurate for the design of a chemical process. 

This new, non-ideal model is only one step removed from an ideal solution. All restrictions remain the same except that the intermolecular forces are no longer uniform (hence G and H will be non-ideal, but S should remain ideal, and V nearly so). The special conditions we have just described define what is called a regular solution. 

---