Gibbs-Duhem-Margules equation

Gibbs-Duhem Equation Gibbs-Duhem-Margules Equation... 

The replacement of all extensive by intensive variables again yields the Gibbs-Duhem-Margules equation. 

Each of the v phases of a heterogeneous system of k components is subjected to a Gibbs-Duhem-Margules equation [see Eq. (27)] ... 

The measurement of osmotic coefficients combined with the Gibbs-Duhem-Margules equation is a well-established method for the determination of the activity coefficients of solutes. 

The integral thermodynamic functions can be calculated with help of Gibbs-Duhem or Gibbs-Duhem- Margules equations ... 

This equation is extremely important (see Section 5.12 for some applications). It is known as the Gibbs-Duhem equation, and such equations as the Duhem-Margules equation may be derived from it. Since no limitation has been put on the type of system considered in the derivation, this equation must be satisfied for every phase in a heterogenous system. We recognize that the convenient independent variables for this equation are the intensive variables the temperature, the pressure, and the chemical potentials. 

In the principal form, corresponding equations also exist for component B. But experimentally, the vapor pressure of only one component, the more volatile component, can be determined. The vapor pressure of the other component can be calculated by the Gibbs-Duhem or the Duhem-Margules equation, as will be explained in detail in Section 3.2.3. 

Figure 3.16 Partial molar Gibbs energy of Au in the alloy Ag u, as function of the mole fraction of Ag. Two sets of values are shown The first set (squares) was calculated with the Gibbs-Duhem equation, as described in Section 3.2.3, and the second set of data (circles) was calculated with the Duhem-Margules equation, temperature 500 °C.

Equation (49a) is the exact differential of quantity Z [Eq. (3)] at constant pressure and temperature. Equation (49b) is a Gibbs-Duhem-Margules type of equation, indicating the mutual dependence of partial molar quantities. 

In the case of a binary system the Gibbs-Duhem (or Duhem-Margules) equation can be put into a useful form in terms of the total pressure. Let x be the mole fraction of component A in the condensed phase and let y be its mole fraction in the vapour phase. It will be assumed that the system contains components A and B only, i.e. there is no inert gas. Then... 

Since the Margules expansions represent a convergent power series in the mole fractions,8 they can be summed selectively to yield closed-form model equations for the adsorbate species activity coefficients. A variety of two-parameter models can be constructed in this way by imposing a constraint on the empirical coefficients in addition to the Gibbs-Duhem equation. For example, a simple interpolation equation that connects the two limiting values of f (f°° at infinite dilution and f = 1.0 in the Reference State) can be derived after imposing the scaling constraint... 

Although the correlations provided by the Margules equations for the two sets of VLE data presented here are satisfactory, they are not perfect. The two possible reasons are, first, that the Margules equations are not precisely suited to the data set second, that the data themselves are systematically in error such that they do not conform to the requirements of the Gibbs/ Duhem equation. 

This is the precise form of what is known as the Duhem>Maigules equation, derived independently, and in various ways, by J. W. Gibbs (1876), P. Duhem (1886), M. Margules (1895), and R. A. Lehfeldt (1895). It is frequently encountered and employed in a less exact form which is based on the approximation that the vapor behaves as an ideal gas. In this event, the fugacity of each component in the vapor may be replaced by its respective partial (vapor) pressure, so that equation (34.12) becomes... 

In this section, we consider a real mixture of two components. We describe Margules expansions for the logarithms of the activity coefficients and investigate relations between the coefficients in the expansions of log/, and log/2 imposed by the Gibbs-Duhem equation. 

Over the range of concentrations between the solubility limits the apparent activity coefficients will vary inversely as the concentrations based on the mixture as a whole. Elimination of 7 s between Eq. (3.79) and any of the integrated Gibbs-Duhem equations therefore permits the estimation of Aab and Aba from the mutual solubility. Thus, the Margules equations lead to... 

The Gibbs-Duhem equation in relation to the Margules and van Laar equationsf... 

The Margules equation is generalized by Gibbs-Duhem equation ... 

Therefore, the two-suffix Margules equation satisfies the Gibbs-Duhem equation since ... 

---