Gibbs-Duhem restriction

The Gibbs-Duhem restriction for constant temperature and pressure can be written for a ternary system as... 

Gibbs-Duhem restriction on the chemical potential (eq. 8.5-3). Eq.(8.5-5) is the generalized Maxwell-Stefan constitutive relation. However, such form is not useful to engineers for analysis purposes. To achieve this, we need to express the chemical potential in terms of mole fractions. This is done by using eq. (8.5-2) into the constitutive flux equation (8.5-5). 

In the previous examples we have assumed that all comonents are present in all of the phases, and we have not introduced any chemical reactions or restrictions. When a component does not exist in a phase, the mole number of that component is zero in that phase and its chemical potential does not appear in the corresponding Gibbs-Duhem equation. As an example, consider a two-component, three-phase system in which two of the phases are pure and the third is a solution. The Gibbs-Duhem equations are then... 

In the second case the liquid and vapor are at equilibrium in a closed vessel without restrictions. An inert gas is pumped into the vessel at constant temperature in order to increase the total pressure. For the present we assume that the inert gas is not soluble in the liquid. (The system is actually a two-component system, but it is preferable to consider the problem in this section.) The Gibbs-Duhem equations are now... 

Both expressions are known as the Gibbs-Duhem relation. Again, these relations impose important constraints, this time on the chemical potentials encountered in a mixture of different components. An example of such a restriction will be furnished in Section 3.14. 

This is the Gibbs/Duhem equation for the adsorbate. Restricting it to constant temperature produces the Gibbs adsorption isotherm ... 

Thus, the interrelationships provided by Eqs. 8.2-8 through 8.2-15 are really restrictions on the mixture equation of state. As such, these equations are important in minimizing the amount of experimental data necessary in evaluating the thermodynamic, properties of mixtures, in simplifying the description of multicomponent systems, and in testing the consistency of certain types of experimental data (see Chapter 10). Later in this chapter we show how the equations of change for mixtures and the Gibbs-Duhem equations provide a basis for the experimental determination of partial molar properties. 

The phase rule can be derived in an alternative fashion from a consideration of the Gibbs-Duhem equations (6-60) for the phases of the heterogeneous system and the restrictions imposed on the variations of the intensive variables and/ = 1,. . . , r a = l,. . . v,... 

A more general version of the Gibbs-Duhem equation, without the restriction of constant T and p, is... 

We first note that mechanical equilibrium imposes a restriction on the driving forces through the Gibbs-Duhem equation cj grad/i, j- = 0. We also note that... 

We study the consequences of applying the Gibbs-Duhem relation to a two-component system at constant T and P. This should make that relationship appear much less abstract it imposes important restrictions on components in equilibrium. We proceed as follows Divide the relation - - U2dfi2 = 0 by (ni + 2) to obtain x dfii + (1 x )dfi2 = 0. With dx2 = —dx we find that... 

This expression represents the Gibbs-Duhem equation and indicates that the intensive properties of the mixture temperature, pressure and partial molar properties, cannot vary independently. Restricted to constant T and P, Eq. 11.6.3 becomes ... 

We will present two applications of the restricted form of the Gibbs-Duhem equation in Examples 11.3 and 11.4, while Eq. 11.6.3 will be used in Chapter 13 to evaluate the thermodynamic consistency of vapor-liquid equilibrium data. 

Introduction of Eqs (A) and (B) into the restricted form of the Gibbs-Duhem equation yields ... 

We conclude, therefore, that the interdependency between the two partial molar volumes, required by the Gibbs-Duhem equation, restricts the values of a and b to zero. Under these conditions, Eqs (A) and (B) refer to an ideal solution. 

---