Perfect binary solution

For a perfect binary solution the free enthalpy (Gibbs energy) of mixing per mole has been given in Eq. 8.7. We extend this equation 8.7 to a non-ideal binary solution by using the activity coefficients and y2 as shown in Eq. 8.17 ... 

This excess volume vE is the difference between the mean molar volume of the non-ideal binary solution, v"° dcal = V+ n2), and the mean molar volume of the perfect binary solution vper/ = XjV + x2v2 (the sum of the volume of the two pure substances before mixing... 

Figure 8.2 The vapor pressure diagram of a perfect binary solution for...

So, for this binary solution of components A and B, which mix perfectly at all compositions, there is a two-phase region at which both solid and liquid phases can coexist. The uppermost boundary between the liquid and liquid + solid phase regions in Figure 2.3f is known as the liquidus, or the point at which solid first begins to form when a melt of constant composition is cooled under equilibrium conditions. Similarly, the lower phase boundary between the solid and liquid + solid phase regions is known as the solidus, or the point at which solidification is complete upon further equilibrium cooling at a fixed composition. 

Experiments show that Raoult s law is obeyed only approximately for a number of binary solutions. It is obeyed perfectly only in case of ideal solutions. So, a solution of two or more components is said to be ideal if it obeys Raoult s law perfectly at all temperatures and concentrations. 

In contrast to a perfect solution, a solution is called an ideal solution, if Eq. 8.1 is valid for solute substances in the range of dilute concentrations only. Moreover, the unitary chemical potential p2(T,p) of solute substance 2 is not the same as the chemical potential p2( T,p) of solute 2 in the pure substance p2(T,p) p2(T,p) Henry s law. For the main constituent solvent, on the other hand, the unitary chemical potential p[( T,p) is normally set to be equal to f l p) in the ideal dilute solution p"(T,p) = p°(l p). The free enthalpy per mole of an ideal binary solution of solvent 1 and solute 2 is thus given by Eq. 8.10 ... 

The enthalpy and volume per mole of a binary solution both vary linearly with the molar fraction x2 of solute 2 in the whole range of x2 for a perfect solution and in a limited dilute range of x2 for a dilute ideal solution, as schematically shown for the volume per mole of a binary solution in Fig. 8.1. 

The difference in thermodynamic functions between a non-ideal solution and a comparative perfect solution is called in general the thermodynamic excess function. In addition to the excess free enthalpy gE, other excess functions may also be defined such as excess entropy sE, excess enthalpy hE, excess volume vE, and excess free energy fE per mole of a non-ideal binary solution. These excess functions can be derived as partial derivatives of the excess free enthalpy gE in the following. 

This difference between perfect solutions and ideal dilute solutions is a consequence of the fact that the enthalpy and volume of mixing, which are both zero for perfect solutions, are not necessarily zero for ideal solutions. For example, if we consider for simplicity a dilute binary solution, the enthalpy before mixing is... 

Fig. 21.1 shows the partial vapour pressures and the total pressure of a perfect solution as a function of composition. In the case of a binary solution (21.8) may be written... 

Raoult s law /rah-oolz/ A relationship between the pressure exerted by the vapor of a solution and the presence of a solute. It states that the partial vapor pressure of a solvent above a solution (p) is proportional to the mole fraction of the solvent in the solution (X) and that the proportionaUty constant is the vapor pressure of pure solvent, (po), at the given temperature i.e. p = PqX. Solutions that obey Raoult s law are said to be ideal. There are some binary solutions for which Raoult s law holds over all values of X for either component. Such solutions are said to be perfect and this behavior occurs when the intermolecular attraction between molecules within one component is almost identical to the attraction of molecules of one component for molecules of the other (e.g. chlorobenzene and bromobenzene). Because of solvation forces this behavior is rare and in general Raoult s law holds only for dilute solutions. 

A binary solution is in equilibrium with its vapour cmd the latter may be assumed to be a perfect gaseous mixture. 

Let us consider for simplification a binary perfect solution consisting of solvent 1 and solute 2. The free enthalpy (Gibbs energy) for one mole of a binary mixture is then given by Eq. 8.6 ... 

Fig. 8.1 Volume per mole as a function of the molar fraction x2 of solute 2 in a binary perfect solution and in an ideal dilute solution v2 = the unitary partial molar volume of solute 2 extrapolated to x2 — 1.

In the intermediate domain of values for the parameters, an exact solution requires the specific inspection of each configuration of the system. It is obvious that such an exact theoretical analysis is impossible, and that it is necessary to dispose of credible procedures for numerical simulation as probes to test the validity of the various inevitable approximations. We summarize, in Section IV.B.l below, the mean-field theories currently used for random binary alloys, and we establish the formalism for them in order to discuss better approximations to the experimental observations. In Section IV.B.2, we apply these theories to the physical systems of our interest 2D excitons in layered crystals, with examples of triplet excitons in the well-known binary system of an isotopically mixed crystal of naphthalene, currently denoted as Nds-Nha. After discussing the drawbacks of treating short-range coulombic excitons in the mean-field scheme at all concentrations (in contrast with the retarded interactions discussed in Section IV.A, which are perfectly adapted to the mean-field treatment), we propose a theory for treating all concentrations, in the scheme of the molecular CPA (MCPA) method using a cell... 

Impurities, i.e. components chemically different from the constituents which form the perfect crystal structure, may be accommodated in the structure, in similar ways to the above, provided that the atoms, ions or molecules of the additive are chemically compatible with those of the host phase. This usually means that their size, shape and bonding properties are similar to those of the constituents of the host material. Incorporation is a specific process. Some binary, or more complex, combinations have only restricted ranges of mutual solubility, while other mixtures are capable of forming the complete range of intermediate solid solutions from 100%Ato 100%B. 

Solid Solutions. An examination of the many published equilibrium diagrams for binary alloy systems shows that generally the addition of one metal to another does not immediately result in the production of a new phase. The normal tendency is, in fact, for a solid solution to form, and it appears at first that the extent of the solid solubility depends on certain characteristics of the metals concerned, but seems to vary in a haphazard way when we pass from one alloy system to another. During the last ten years much research work has been carried out on the problems of solid solution and solid solubility, and we are now in a position to discuss the results in some detail. It has been shown that formation (or otherwise) of solid solution and extent of solid solubility can, in most cases, be perfectly satisfactorily explained. 

Although the Murphree model contains an additional assumption (that the liquid leaving plate j is at its bubble-point temperature) over the modified Murphree model, the corresponding values of y 1 predicted by both models on the basis of the same sets coefficients n Li, nGi and points xylf yj+ lf i appear to be in almost perfect agreement for the two examples presented (see Tables 13-2 and 13-3). These examples were taken from Ref. 24. The number of transfer units for each film in these examples was taken to be independent of component identity just as they are for the existing correlations for binary mixtures which are given below. In Example 13-1 (the benzene-toluene system), the vapor and liquid phases closely approximate ideal solutions, but the liquid phase of the ethanol-water system in Example 13-2 is highly nonideal. 

It is important to note here that the solution of one compound in another is unavoidable — a perfectly pure crystal is a thermodynamic impossibility for the same reason that a defect-free crystal is impossible. " The only legitimate question therefore is. How much solubility is there In many binary systems, the regions of solid solution that are necessarily present do not appear on the phase diagrams. For example, according to Fig. 8.7 or... 

Calculation of the degree of separation of a binary mixture in a membrane module for cocurrent or countercurrent flow patterns involves the numerical solution of a system of two nonlinear, coupled, ordinary differential equations (Walawender and Stem, 1972). For a given cut, the best separation is achieved with countercurrent flow, followed by crossflow, cocurrent flow, and perfect mixing, in that order. The crossflow case is considered to be a good, conservative estimate of module membrane performance (Seader and Henley, 2006). 

For a binary system in which a molecular species A constitutes the solvent and a species B the solute, there is an important result, of which use has already been made in anticipation, namely that in the limit of sufficiently small concentrations % = c. This may be expressed by saying that in dilute enough solutions B follows the perfect gas laws. This statement is worthy of some detailed considera-... 

Consider convection diffusion toward a spherical particle of radius R, which undergoes translational motion with constant velocity U in a binary infinite diluted solution [3], Assume the particle is small enough so that the Reynolds number is Re = UR/v 1. Then the flow in the vicinity of the particle will be Stoke-sean and there will be no viscous boundary layer at the particle surface. The Peclet diffusion number is equal to Peo = Re Sc. Since for infinite diluted solutions, Sc 10 and the flow can be described as Stokesian for the Re up to Re 0.5, it is perfectly safe to assume Pec 1. Thus, a thin diffusion boundary layer exists at the surface. Assume that a fast heterogeneous reaction happens at the particle surface, i.e. the particle is dissolving in the liquid. The equation of convective diffusion in the boundary diffusion layer, in a spherical system of coordinates r, 6, (p, subject to the condition that concentration does not depend on the azimuthal angle [Pg.128]

Secondary thermodynamic tables are unfortunately much more sparse. For example, while the deviations from perfect-gas behaviour have been studied for many pure gases, relatively little information is yet available even for binary gas mixtures, let alone for the usually multi-component mixtures relevant to chemical reactions in gases. Again, while the activity coefficient of the solute or the osmotic coefficient of the solvent has been measured over useful molality ranges for many solutions of a single electrolyte, little such information is yet available for the mixed electrolyte solutions relevant to chemical reactions in solutions. 

Out of the three types of interaction parameters, it is almost exclusively x that is of relevance for the thermodynamic description of binary and ternary polymer-containing liquids, as will be described in the section on experimental methods (Sect. 3). The integral interaction g parameter is practically inaccessible, and the parameter referring to the polymer, suffers from the difficulties associated with the formation of perfect polymer crystals, because it is based on their equilibria with saturated polymer solutions. 

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