Activity coefficient relationships Gibbs-Duhem equation

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-... 

In order to calculate the distribution coefficient by Equation 1.29, the activity coefficient Y must be evaluated. The activity coefficients are generally determined from the experimental data and correlated on the basis of thermodynamic phase equilibrium principles. The relationship most often used for this purpose is the Gibbs-Duhem equation (Equation 1.7). At constant temperature and pressure, this equation becomes... 

The general principles established for ideal solutions, such as Raoult s law in its various forms, are of course applicable to solutions of any number of components. Similarly, the Gibbs-Duhem equation is applicable to nonideal solutions of any number of components, and as in the case of binary mixtures various relationships can be worked out relating the activity coefficients for ternary mixtures. This problem has now been attacked from several points of view, a most excellent summary of which is presented by Wolil (35). His most important results pertinent to the problem at hand are summarized here. 

But we have also seen in Section 13.9, that the two activity coefficients are interrelated through the Gibbs-Duhem equation, thus providing an additional equation. We examine next how this relationship is used in determining yi. 

In this section, we explore the relationship between the activity coefficients of the different species in a mixture. Using the Gibbs-Duhem equation, we will show that the activity coefficients are not independent. Their interrelationship will motivate development of a new type of thermodynamic property—excess Gibbs energy. Finally, we will illustrate an application of these principles by coming up with a way to test the quality of experimental data and see whether they are thermodynamically consistent. 

In Section 6.3, we used the Gibbs—Duhem equation to provide a relationship between the partial molar properties of different species in a mixture. We can use this equation to relate the activity coefficients of different species in a mixture as well. We begin by writing Equation (6.19) in terms of partial molar Gibbs energy, that is, chemical potential ... 

Equation (7.43) says that the activity coefficients of different species in the mixture are interrelated. Thus, when we develop models to fit activity coefficients in the next section, we must make sure that the expressions for the activity coefficients of different species in a mixture are consistent with the Gibbs—Duhem equation. On the other hand, this relationship also suggests an intriguing possibility— perhaps it is possible to consolidate the activity coefficient dependence on composition of all the species into one model expression and, therefore, be able to derive all the activity coefficients from a single expression. Indeed, as we will discover next, we will take just this approach through a new thermodynamic property—the excess Gibbs energy. 

---