Nonideal solutions Gibbs energy

Solntions in which the concentration dependence of chemical potential obeys Eq. (3.6), as in the case of ideal gases, have been called ideal solutions. In nonideal solntions (or in other systems of variable composition) the concentration dependence of chemical potential is more complicated. In phases of variable composition, the valnes of the Gibbs energy are determined by the eqnation... 

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. 

When no distinction between the solvent and the solute in a liquid mixture is made, then nonideal mixtures can still be described by means of an expression similar to Eq. (2.16), but with the addition of a term that is the excess molar Gibbs energy of mixing, AGab (see also section 2.2). Thus the Gibbs energy per mole of mixture is ... 

FIGURE 3.13.1 The Gibbs free energy of mixing for a nonideal solution according to Eq. (3.13.13), for various values of the parameter B. 

In order to perform quantitative thermodynamic calculations using the Gibbs free energy for a nonideal solution (see Eqs. (6.9)—(6.11)), we need explicit expressions for the activity coefficients. A few empirical expressions that are typically employed are ... 

The excess Gibbs energy is the sum of the nonidealities of the components and is the measure of the departure of the total solution from ideal-solution behavior. Hence it is the focal point for theoretical formulation. To facilitate the formulation, g is conveniently separated into parts. 

The ideal solution assumes equal strength of self- and cross-interactions between components. When this is not the case, the solution deviates from ideal behavior. Deviations are simple to detect upon mixing, nonideal solutions exhibit volume changes (expansion or contraction) and exhibit heat effects that can be measured. Such deviations are quantified via the excess properties. An important new property that we encounter in this chapter is the activity coefficient. It is related to the excess Gibbs free energy and is central to the calculation of the phase diagram. 

Several equations exist for the activity coefficients in nonideal solution. These are based on models for the excess Gibbs free energy of the solution and they are known by the name of the scientists who developed them or by the theory used to model nonideal interactions. The literature on this topic is extensive and remains the subject of research. Here we review some of the most common models that have been used to relate the activity coefficient to composition. 

At this point we have a fundamental problem. Given the relationship between Gibbs energies and compositions for ideal solutions we have developed, how do we handle deviations from this behavior What mathematical form should our equations for nonideality take There is a variety of approaches for this. The most general is to develop an equation of state, and there is a variety of types of those (Chapter 13). Then there are different approaches for dilute and concentrated solutions, and for electrolytes and nonelectrolytes. In this section we look at some fairly general methods which have been applied to many solid and liquid solutions. 

For nonideal solutions, the molar Gibbs free energy of mixing is... 

The difference between the Gibbs free energies of mixing of ideal and nonideal solutions is called the excess Gibbs free energy, which we shall denote by AGe. From (8.4.7) and (8.4.16) it follows that... 

The thermodynamic functions of solutions are generally expressed in terms of the changes produced by mixing the pure components to form the solution at constant temperature and pressure. From Euler s theorem and Eq. (6.3-6), the Gibbs energy of a nonideal solution is given by... 

The first sum in the right-hand side of the final version of this equation is the same as for an ideal solution, and the second sum represents a correction for the nonideality of the solution. This contribution is called the excess Gibbs energy and is denoted by G ... 

Physical Equilibria and Solvent Selection. In order for two separate Hquid phases to exist in equiHbrium, there must be a considerable degree of thermodynamically nonideal behavior. If the Gibbs free energy, G, of a mixture of two solutions exceeds the energies of the initial solutions, mixing does not occur and the system remains in two phases. Eor the binary system containing only components A and B, the condition (22) for the formation of two phases is... 

A number of attempts have been made to use the simple solution model to represent the solution nonidealities in binary (23,29-32) and ternary (23,33-41) III—V systems. In the simple solution model, the integral Gibbs excess energy is given in terms of an interaction energy a)( T) by the equation... 

As with other nonideal chemical systems, in order to better understand ionic solutions we will go back to the concepts of chemical potentials and activities. In Chapter 4, we defined the chemical potential yu, of a material as the change in the Gibbs free energy versus the molar amount of that material ... 

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