Ideal solutions and

There is a parallel between the composition of a copolymer produced from a certain feed and the composition of a vapor in equilibrium with a two-component liquid mixture. The following example illustrates this parallel when the liquid mixture is an ideal solution and the vapor is an ideal gas. 

We define Fj to be the mole fraction of component 1 in the vapor phase and fi to be its mole fraction in the liquid solution. Here pj and p2 are the vapor pressures of components 1 and 2 in equihbrium with an ideal solution and Pi° and p2° are the vapor pressures of the two pure liquids. By Dalton s law, Plot Pi P2 Pi/Ptot these are ideal gases and p is propor-... 

Evaluate ASj for ideal solutions and for athermal solutions of polymers having n values of 50, 100, and 500 by solving Eqs. (8.28) and (8.38) at regular intervals of mole fraction. Compare these calculated quantities by preparing a suitable plot of the results. 

Here d is the disparity between the free energy per ion pair added to the non-ideal solution and the free energy per ion pair added to the corresponding ideal solution. It is the disparity between the communal term in the free energy and the cratic term in the free energy. In the solution... 

Turning now to the non-ideal solution, we may answer question (1) by saying that the value of (163) will vary with concentration only insofar as the solution differs from an ideal solution and we can proceed to ask a third question how would the value of (163) vary with concentration for an ionic solution in the extremely dilute range We must answer that in a series of extremely dilute solutions the value of (163) would be constant within the experimental error it is, in fact, a unitary quantity, characteristic of the solute dissolving in the given solvent. As in See. 55, this constant value adopted by (163) in extremely dilute solutions may conveniently be written as the limiting value as x tends to zero thus... 

The equality of activity and mole fraction in an ideal solution has interesting consequences. These consequences, in fact, are the characteristics of an ideal solution, and are presented in the following. 

The zinc will be inclined to depart spontaneously from the high-activity amalgam to that with a corresponding low activity. For example, if c1 is greater than c2, E is positive and the reaction advances in the direction specified. It may be added that metals in mercury constitute fairly ideal solutions, and that the emfs are almost correctly calculated by using concentrations instead of activities. 

For a solution of a non-volatile substance (e.g. a solid) in a liquid the vapour pressure of the solute can be neglected. The reference state for such a substance is usually its very dilute solution—in the limiting case an infinitely dilute solution—which has identical properties with an ideal solution and is thus useful, especially for introducing activity coefficients (see Sections 1.1.4 and 1.3). The standard chemical potential of such a solute is defined as... 

Fig. 10 Dependence of vapor pressure of a solution containing a volatile solute, illustrated for (A) an ideal solution and (B) a nonideal solution and shown as a function of mole fraction of the solute. Individual vapor pressure curves are shown for the solvent (0) the solute ( ), and for the sum of these (X).

The most common model for describing adsorption equilibrium in multi-component systems is the Ideal Adsorbed Solution (IAS) model, which was originally developed by Radke and Prausnitz [94]. This model relies on the assumption that the adsorbed phase forms an ideal solution and hence the name IAS model has been adopted. The following is a summary of the main equations and assumptions of this model (Eqs. 22-29). 

This evaluation assumes an ideal solution and the formation of a 1 1 complex. 

For solutions obeying Henry s law, as for ideal solutions, and for solutions of ideal gases, the chemical potential is a linear function of the logarithm of the composition variable, and the standard chemical potential depends on the choice of composition variable. The chemical potential is, of course, independent of our choice of standard state and composition measure. 

Note 1 In some respects, a polymer solution in the theta state resembles an ideal solution and the theta state may he referred to as a pseudo-ideal state. However, a solution in the theta state must not be identified with an ideal solution. 

Figure 3-36 The dependence of interdiffusivity on composition for two models (Equations 3-137c versus 3-138b) for ideal solutions and concentration-independent T>a and X>b- The solid curve is for interdiffusion of two ions of identical charge. The dashed curve is for interdiffusion of neutral atomic species such as in an alloy.

In support of the conclusion based on silver, series of 0.2, 0.5, 1.0, 2.0, and 5.0 % w/w of platinum, iridium, and Pt-Ir bimetallic catalysts were prepared on alumina by the HTAD process. XRD analysis of these materials showed no reflections for the metals or their oxides. These data suggest that compositions of this type may be generally useful for the preparation of metal supported oxidation catalysts where dispersion and dispersion maintenance is important. That the metal component is accessible for catalysis was demonstrated by the observation that they were all facile dehydrogenation catalysts for methylcyclohexane, without hydrogenolysis. It is speculated that the aerosol technique may permit the direct, general synthesis of bimetallic, alloy catalysts not otherwise possible to synthesize. This is due to the fact that the precursors are ideal solutions and the synthesis time is around 3 seconds in the heated zone. 

Fig. 10. Composition fluctuation factor at zero wave number for Ga-Sb melt at 709.2°C. Curve 1 is for the parameters established here, whereas curve 2 is for a completely dissociated, ideal solution and curve 3 for a completely associated ideal solution. 

Our approach has been essentially empirical in nature with less emphasis on the theoretical. We have isolated single substances, proved their purity, and determined their covalent structure by classical methods of organic chemistry we have then used these substances of molecular weight ranging from 1,000 to 14,000 as model solutes for the study of conformation and intermolecular interaction. Solutes of special interest have been gramicidin SA (2), bacitracin A (3), polymyxin B, and the tyrocidines A, B, and C (4). All are cyclic antibiotic polypeptides. The first three behave in aqueous solution as reasonably ideal solutes and do not associate, but the tyrocidines associate strongly and are interesting models for the study of association phenomena. Other model solutes of... 

Assuming an ideal solution and a high temperature, the following equations are valid ... 

Step 1. For a function to be minimized, determine its ideal solution and anti-ideal solutions by directly minimizing and maximizing the objective function. 

In the following sections expressions are developed for the chemical potential of the components or species in solution in terms of the composition —first for an ideal solution and then for real solutions—with special emphasis on reference and standard states. 

For exact calculations, values of the excess chemical potentials of all the species must be known or calculable on some theoretical basis. Unfortunately, this is not generally the case. Under such circumstances it is convenient to assume that the species form an ideal solution and that all deviations based... 

Assume that benzene and naphthalene form an ideal solution and that the solids are pure components (no solid solutions are formed) with melting points at 1.0 atm pressure of 5.5°C and 80.5°C. Estimate the composition and melting point of the benzene-naphthalene eutectic using Eq. (77). 

A solution is defined as a condensed phase (liquid or solid) containing several substances. The main substance of the solution is called solvent and the other constituent substances dissolved in the solvent are solutes. Solutions are classified into ideal solutions and non-ideal solutions. For an ideal solution the chemical potential of a constituent substance i is given by ... 

We first take as a reference system an infinitely dilute solution of solute 2 in solvent 1. The chemical potentials of solvent 1 and solute 2, then, are given in the form of Eq. 8.13 for an ideal solution and in the form of Eq. 8.14 for a non-ideal solution ... 

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. 

It follows from the last equation that the net work gained by the dilution of an actual solution of an ion concentration c (c > 1) to the state of an ideal solution, with concentration c = 1, equals the sum of two constituents. The term of the first constituent, RT In c, expresses the energy gained by the dilution of the ideal solution and the second one, RT In y, represents the work, required to overcome the effect of the interionic electrical forces. Since the last mentioned electrical work reduces tho value of the work W, which would l>e gained by the dilution of an ideal solution, the expression RT In y must have a negative value, or in other words the activity coefficient of actual solutions y < 1. It follows further from this conception that the potential of an electrolyte contained in an actual solution must be lower than the potential of the same substance in an ideal solution. 

The properties of mixtures of ideal gases and of ideal solutions depend solely on the properties of the pure constituent species, and are calculated from them by simple equations, as illustrated in Chap. 10. Although these models approximate the behavior of certain fluid mixtures, they do not adequately represent the -behavior of most solutions of interest to chemical engineers, and Raoult s law is not in general a realistic relation for vapor/liquid equilibrium. However, these models of ideal behavior—the ideal gas, the ideal solution, and Raoult s law— provide convenient references to which the behavior of nonideal solutions may be compared. 

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