Gibbs-Duhem equation activity

It is strictly for convenience that certain conventions have been adopted in the choice of a standard-state fugacity. These conventions, in turn, result from two important considerations (a) the necessity for an unambiguous thermodynamic treatment of noncondensable components in liquid solutions, and (b) the relation between activity coefficients given by the Gibbs-Duhem equation. The first of these considerations leads to a normalization for activity coefficients for nonoondensable components which is different from that used for condensable components, and the second leads to the definition and use of adjusted or pressure-independent activity coefficients. These considerations and their consequences are discussed in the following paragraphs. 

Equation (16) is a differential equation and applies equally to activity coefficients normalized by the symmetric or unsymme-tric convention. It is only in the integrated form of the Gibbs-Duhem equation that the type of normalization enters as a boundary condition. 

If we vary the composition of a liquid mixture over all possible composition values at constant temperature, the equilibrium pressure does not remain constant. Therefore, if integrated forms of the Gibbs-Duhem equation [Equation (16)] are used to correlate isothermal activity coefficient data, it is necessary that all activity coefficients be evaluated at the same pressure. Unfortunately, however, experimentally obtained isothermal activity coefficients are not all at the same pressure and therefore they must be corrected from the experimental total pressure P to the same (arbitrary) reference pressure designated P. This may be done by the rigorous thermodynamic relation at constant temperature and composition ... 

The Gibbs-Duhem equation allows the determination of activity coefficients for one component from data for those of other components. 

The Gibbs-Duhem equation is extremely important in solution chemistry and it can be seen from equation 20.171 that it provides a means of determining the activity of one component in a binary solution providing the activity of the other is known. 

Trustworthy thermodynamic data for metal solutions have been very scarce until recently,25 and even now they are accumulating only slowly because of the severe experimental difficulties associated with their measurement. Thermodynamic activities of the component of a metallic solution may be measured by high-temperature galvanic cells,44 by the measurement of the vapor pressure of the individual components, or by equilibration of the metal system with a mixture of gases able to interact with one of the components in the metal.26 Usually, the activity of only one of the components in a binary metallic solution can be directly measured the activity of the other is calculated via the Gibbs-Duhem equation if the activity of the first has been measured over a sufficiently extensive range of composition. 

B. Constant-Pressure Activity Coefficients and the Gibbs-Duhem Equation.. 158... 

The advantages of constant-pressure activity coefficients also become clear when we try to relate to one another the activity coefficients of all the components in a mixture through the Gibbs-Duhem equation (P6, P7). For variable-pressure activity coefficients at constant temperature we obtain... 

Most of the methods we have described so far give the activity of the solvent. Often the activity of the solute is of equal or greater importance. This is especially true of electrolyte solutions where the activity of the ionic solute is of primary interest, and in Chapter 9, we will describe methods that employ electrochemical cells to obtain ionic activities directly. We will conclude this chapter with a discussion of methods based on the Gibbs-Duhem equation that allow one to calculate activities of one component if the activities of the other are known as a function of composition. 

The osmotic coefficient 4> and activity coefficient are related in a simple manner through the Gibbs-Duhem equation. We can find the relationship by writing this equation in a form that relates a and 2-... 

To do so we need only specify a dilution sufficient to assure that the activity of component 1 is proportional to its concentration. Application of the Gibbs-Duhem equation leads to Eq. (30). 

The thermodynamic properties of real electrolyte solutions can be described by various parameters the solvent s activity Oq, the solute s activity the mean ion activities a+, as well as the corresponding activity coefficients. Two approaches exist for determining the activity of an electrolyte in solution (1) by measuring the solvent s activity and subsequently converting it to electrolyte activity via the thermodynamic Gibbs-Duhem equation, which for binary solutions can be written as... 

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. 

Thermodynamic methods also measure the activity coefficient of the solvent (it should be recalled that the activity coefficient of the solvent is directly related to the osmotic coefficient—Eq. 1.1.19). As the activities of the components of a solution are related by the Gibbs-Duhem equation, the measured activity coefficient of the solvent can readily be used to calculate the activity coefficient of a dissolved electrolyte. 

Many reactions encountered in extractive metallurgy involve dilute solutions of one or a number of impurities in the metal, and sometimes the slag phase. Dilute solutions of less than a few atomic per cent content of the impurity usually conform to Henry s law, according to which the activity coefficient of the solute can be taken as constant. However in the complex solutions which usually occur in these reactions, the interactions of the solutes with one another and with the solvent metal change the values of the solute activity coefficients. There are some approximate procedures to make the interaction coefficients in multicomponent liquids calculable using data drawn from binary data. The simplest form of this procedure is the use of the equation deduced by Darken (1950), as a solution of the ternary Gibbs-Duhem equation for a regular ternary solution, A-B-S, where A-B is the binary solvent... 

In experimental investigations of thermodynamic properties of solutions, it is common that one obtains the activity of only one of the components. This is in particular the case when one of the components constitutes nearly the complete vapour above a solid or liquid solution. A second example is when the activity of one of the components is measured by an electrochemical method. In these cases we can use the Gibbs-Duhem equation to find the activity of the second component. 

A graphical integration of the Gibbs-Duhem equation is not necessary if an analytical expression for the partial properties of mixing is known. Let us assume that we have a dilute solution that can be described using the activity coefficient at infinite dilution and the self-interaction coefficients introduced in eq. (3.64). 

By examining the compositional dependence of the equilibrium constant, the provisional thermodynamic properties of the solid solutions can be determined. Activity coefficients for solid phase components may be derived from an application of the Gibbs-Duhem equation to the measured compositional dependence of the equilibrium constant in binary solid solutions (10). 

The activity of the water is derived from this expression by use of the Gibbs-Duhem equation. To utilize this equation, the interaction parameters fif ) and BH must be estimated for moleculemolecule, molecule-ion and ion-ion interactions. Again the method of Bromley was used for this purpose. Fugacity coefficienls for the vapor phase were determined by the method of Nakamura et al. (JO). 

The activity of the solvent (water) in a solution of pure electrolyte dissolved in water can be computed by application of the Gibbs-Duhem equation ... 

Equations 8 and 9 can be used with the Gibbs-Duhem equation to calculate pf,., the activity coefficient of water, for each of the binary systems. The Gibbs-Duhem equation for a binary aqueous electrolyte solution is written ... 

For the ternary solution, the Gibbs-Duhem equation can be easily integrated to calculate the activity coefficient of water when the expressions for the activity coefficients of the electrolytes are written at constant molality. For Harned s rule, integration of the Gibbs-Duhem equation gives the activity of water as ... 

Equations 11 and 12 are not written for constant molality, and can not be easily used with the Gibbs-Duhem equation to obtain an analytical expression for the activity of water in the ternary solution. However, it is possible to propose a separate equation for the activity coefficient of water that is consistent with the proposed model of concentrated solutions. 

In Chapters 16 and 17, we developed procedures for defining standard states for nonelectrolyte solutes and for determining the numeric values of the corresponding activities and activity coefficients from experimental measurements. The activity of the solute is defined by Equation (16.1) and by either Equation (16.3) or Equation (16.4) for the hypothetical unit mole fraction standard state (X2° = 1) or the hypothetical 1-molal standard state (m = 1), respectively. The activity of the solute is obtained from the activity of the solvent by use of the Gibbs-Duhem equation, as in Section 17.5. When the solute activity is plotted against the appropriate composition variable, the portion of the resulting curve in the dilute region in which the solute follows Henry s law is extrapolated to X2 = 1 or (m2/m°) = 1 to find the standard state. 

As we saw in Section 17.5, the activity coefficient of a nonelectrolyte solute can be calculated from the activity coefficient of the solvent, which, in turn, can be obtained from the measurement of colligative properties such as vapor pressure lowering, freezing point depression, or osmotic pressure. We used the Gibbs-Duhem equation in the form [Equation (17.33)]... 

Once values of g as a function of solution composition have been obtained, the Gibbs-Duhem equation can be used to relate the osmotic coefficient of the solvent to the activity coefficient of the solute. For this purpose, the chemical potential of the solvent is expressed as in Equation (19.42), with the approximation given in Equation (19.53), so that... 

Using the data of Fig. 38 and data obtained by attempting to duplicate the run, Allen and her coworkers determined the activity coefficients presented in Fig. 39. The two sets of data arc in quite good agreement except at lower mole fractions of DBO, which correspond to the later phases of a run when contamination became significant. Since the activity coefficients for each of the two species were determined from the data, the consistency of the results can be tested by applying the Gibbs-Duhem equation,... 

Fig. 39. Activity coefficients calculated from the data of Fig. 37 and from an additional set of data obtained from a duplicate run compared with the Gibbs-Duhem equation. Reprinted with permission from Allen, T. M., Taflin, D. C, and Davis, E. J., Ind. Eng. Chem. Res. 29, 682-690, Copyright 1990, American Chemical Society.

If we apply Gibbs-Duhem equations (cf section 2.11) we may deduce that the activity of the dissolved basic oxide is entirely represented by the activity of free oxygen in the system, so that... 

An alternative approach is to estimate activity coefficients of the solvents from experimental data and correlate these coefficients using, for example, the Wilson equation. Rousseau et al. (3) and Jaques and Furter (4) have used the Wilson equation, as well as other integrated forms of the Gibbs-Duhem equation, to show the utility of this approach. These authors found it necessary, however, to modify the definitions of the solvent reference states so that the results could be normalized. 

The activities a and a% are related through the Gibbs-Duhem equation ... 

Here, we provide an independent demonstration that activity coefficients obeying Eq. (12) satisfy the Gibbs-Duhem equation. 

Now, let s consider the other form of the Gibbs-Duhem equation which involves activity coefficients in place of activities ... 

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