Thermodynamics of Ideal Solutions

The thermodynamic equations for ideal solutions are derived by considering the equilibrium between a given component in the vapor phase and liquid solution. Thus, for component A in a solution containing two components, 

The thermodynamic condition for equilibrium is that the chemical potential of A in the liquid phase be equal to that of A in the vapor, that is. 

pX° is the standard chemical potential of component A in the liquid solution, which can be measured when the mole fraction of A is one (pure A). A similar analysis for the other component B leads to the equation 

The thermodynamic quantities associated with the mixing of pure liquids to form a solution are important in assessing solution properties. Suppose a moles of component A are combined with b moles of component B to form a solution. The Gibbs energy change associated with this process is given by 

Dividing both sides by the total number of moles, a + b, one obtains 

Despite the dissimilarities in these hypothetical models of ideal gaseous, liquid, and solid solutions, we see that they share a number of important properties. All ideal solutions, no matter what the phase, have no heat of mixing when prepared from their components, and total volumes must simply be the sum of the individual volumes of the components before mixing. 

These relationships, and others, follow directly from the relationship between the partial molar free energy (chemical potential) of a constituent and its mole fraction in an ideal solution 

From (9.10), the molar free energy of the solution is the weighted sum of the chemical potentials of all components, i.e. 

This allows us to break equation (10.2) into two very important parts. One part describes the free energy contribution due solely to the mechanical mixture or sum of free energies of the substances being mixed. This is the first term on the right side 

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Based on the thermodynamics of ideal solutions, it can be shown that for monodispersed solutes ... 

If the adsorption value is small compared to T 10"5 mol m 2 (see Chapter II, 1), the surfactant concentration is low, not only in the bulk, but also within the surface layer. In this case the thermodynamics of ideal solutions can be used to describe both the bulk solution and the surface layer. The chemical potential of the surfactant molecules in the bulk can then be written as... 

This is the symmetrical ideal solution. We shall discuss the thermodynamics of ideal solutions further in Chapter 6. The average number of As (or Bs) in the system is obtained from... 

The steel will be considered to be an ideal ternary solution, and therefore at all temperatures a, = 0-18, Ani = 0-08 and flpc = 0-74. Owing to the Y-phase stabilisation of iron by the nickel addition it will be assumed that the steel, at equilibrium, is austenitic at all temperatures, and the thermodynamics of dilute solutions of carbon in y iron only are considered. 

Figure 14.2. Thermodynamics of formation of ideal solution from pure components.

That surfactant molecules form aggregates designed to remove unfavourable hydrocarbon-water contact is not surprising but the question that should be asked is why the aggregates form sharply at a concentration characteristic of the surfactant (the cmc). From the basic equation of ideal solution thermodynamics... 

Defect thermodynamics, as outlined in this chapter, is to a large extent thermodynamics of dilute solutions. In this situation, the theoretical calculation of individual defect energies and defect entropies can be helpful. Numerical methods for their calculation are available, see [A. R. Allnatt, A. B. Lidiard (1993)]. If point defects interact, idealized models are necessary in order to find the relations between defect concentrations and thermodynamic variables, in particular the component potentials. We have briefly discussed the ideal pair (cluster) approach and its phenomenological extension by a series expansion formalism, which corresponds to the virial coefficient expansion for gases. 

Thermodynamic Properties of Ideal Solutions Equation (11.79) is the starting point for deriving equations for AmjxZ, the change in Zm for forming an ideal mixture.hh For the ideal solution, 7ri(- = 1 and equation (11.79) becomes... 

We have seen that the properties of non-ideal or real solutions differ from those of ideal solutions. In order to consider the deviation from ideality, we may divide thermodynamic mixing properties into two parts ... 

To relate these thermodynamic quantities to molecular properties and interactions, we need to consider the statistical thermodynamics of ideal gases and ideal solutions. A detailed discussion is beyond the scope of this review. We note for completeness, however, that a full treatment of the free energy of solvation should include the changes in the rotational and vibrational partition functions for the solute as it passes from the gas phase into solution, AGjnt. ... 

We now turn to a different class of ideal solutions which has been of central importance in the study of solvation thermodynamics. We shall refer to a dilute ideal (DI) solution whenever one of the components is very dilute in the solvent. The term very dilute depends on the system under consideration, and we shall define it more precisely in what follows. The solvent may be... 

In the following sections we will quantify some of the thermodynamic properties of mechanical mixtures and ideal and non-ideal solutions. As we detail the properties of ideal solutions, it will become clear that they are strictly hypothetical another thermodynamic concept, like true equilibrium , which is a limiting state for real systems. Ideal solutions, in other words, are another part of the thermodynamic model, not of reality. It is a useful concept, because real solutions can be compared to the hypothetical ideal solution and any differences described by using correction factors (activity coefficients) in the equations describing ideal behavior. These correction factors can either be estimated theoretically or determined by actually measuring the difference between the predicted (ideal) and actual behavior of real solutions. 

The problem of hypothetical states also causes difficulties in formulating thermodynamic equations for the density and enthalpy of ideal solutions. However, for low to moderate pressures, the fugacity expressions used in formulating the ideal K-value given by (4-92) can also be used to derive consistent expressions for the other thermodynamic quantities. Equation (4-54), for example, provides a way to obtain pure vapor molal volumes... 

In this chapter, we apply some of the general principles developed heretofore to a study of the bulk thermodynamic properties of nonelectrolyte solutions. In Sec. 11-1 we discuss conventions for the description of chemical potentials in nonelectrolyte solutions and introduce the concept of an ideal component. In Sec. 11-2, we demonstrate how the concept of solution molecular weight can be introduced into thermodynamics in a natural fashion. Section 11-3 is devoted to a study of the properties of ideal solutions. In Sec. 11-4, we discuss the properties of solutions that can be considered to be ideal when they are dilute but are not necessarily ideal when they are more concentrated. In Sec. 11-5, regular solutions are defined and some of their properties are derived. Section 11-6 is devoted to a study of some of the approximations that prove useful in the derivation of the properties of real solutions. Finally, in Sec. 11-7, some of the experimental techniques utilized for the measurement of chemical potentials and activity coefficients of components in solution are described. 

The thermodynamic behavior of real solutions, such as those in which most reactions take place, is based on a description of ideal solutions. The model of an ideal solution is based on Raoult s law. While we can measure the concentration of a species in solution by its mole fraction, X the fact that the solution is not ideal tells us that thermodynamic behavior must be based on fiigacity, fi. In this development, we will use L as the fugacity of the pure component i and f, as the fugacity of component i in the solution. When Xi approaches unity, its fugacity is given by... 

The linear form (5.1.2) is the simplest expression that can be devised for the composition dependence of a fugacity, and in fact (5.1.2) can be considered to be a thermodynamic definition of ideal solution. Even the ideal-gas mixture, for which (ffj = 1, is a special kind of ideal solution that is, the ideal-gas fugacity takes the form (5.1.2) with... 

Since the ideal-solution concept is not restricted to a particular kind of intermolecular force, we have significant flexibility in performing thermodynamic analyses. In many situations, use of one kind of ideal solution may simplify an analysis more than another. For example, calculations are often easier when we use one ideality for nonelectrolyte solutions, another for dilute solutions, another for electrolytes, and yet another for polymeric blends. This degree of flexibility is not obtained by basing aU analyses on ideal gases. 

Relationship (3.7.33) suggests that we use the hard-point particle as our test solute to compare the solvation thermodynamics of this solute in different solvents. We immediately see from (3.7.33) that our test solute will be more soluble in a liquid for which the quantity pV is smaller. In other words, decreasing either the density or the size of the solvent particles causes a decrease in AG or an increase in solubility. Here, we refer to solubility from an ideal gas phase. For real solutes, the attractive... 

As we have learned in Section 1.8, there are a few concentration regimes in the polymer solution. Chapter 2 will primarily focus on the thermodynamics of dilute solutions, that is, below the overlap concentration, although we will also look at how the thermodynamics of the solution deviates from that of the ideal solution with an increasing concentration. Properties characteristic of nondilute solutions will be examined in detail in Chapter 4. 

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