Excess properties, Gibbs-Duhem equation

The Gibbs-Duhem equation (Eq. (4.36)) may be expressed as well for the excess properties ... 

We have seen in Chapter 11 that all partial molar properties must satisfy the Gibbs-Duhem equation, Eq. 11.6.3, which when applied to the molar excess Gibbs free energy yields (Problem 13.52) ... 

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. 

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. 

---