Duhem-Margules equation

Equation (6.77) is known as the Duhem-Margules equation. It can also be written as... 

When deviations from ideal solution behavior occur, the changes in the deviations with mole fraction for the two components are not independent, and the Duhem-Margules equation can be used to obtain this relationship. The allowed combinations"1 are shown in Figure 6.10 in which p /p and P2//>2 are... 

Figure 6.10 Representative deviations from ideal solution behavior allowed by the Duhem-Margules equation. The dotted lines are the ideal solution predictions. The dashed lines giveP2IP2 (lower left to upper right), and p jp (upper left to lower right).

From the Duhem-Margules equation [equation (6.77)], we know that... 

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. 

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

On the basis of the Duhem-Margules equation, prove that if one component of a binary mixture exhibits positive (negative) deviations from Raoult s Law, the second must do likewise. (See S. Glasstone, "Thermodynamics for Chemists", D. Van Nostrand, New York, 1947, Chapter 14.)... 

Duhem-Margules equation for compounds with a small deviation from the stoichiometric composition [ Schottky, Ulich, and Wagner, Thermodynamik, p. 375, Berlin 1929), the sum of the chemical potentials of anions and cations is constant in the crystal (and in its surface), (d) In the steady state of the dissolution process, equivalent numbers of anions and cations have to pass through the phase boundary per unit time. 

The Duhem-Margules equation (9 et seq.) was first confirmed quantitatively by v. Zawidzki. Margules showed that from the relationships p = I>a+P,=Mx)+Ma )... 

The Duhem-Margules Equation and Beatty and Callingaert s Criterion. 

Strictly speaking the partial pressures depend on the total pressure, but it is readily shown, as in 1, that the variation is negligible at ordinary pressure because of the small molar volumes of condensed phases. Equation (21.53) therefore reduces to the Duhem-Margules equation, ... 

By combining the Duhem-Margules equation (21.54) with the equality... 

An increasing amount of attention has been given recently to the problem of manipulating the Duhem-Margules equation in the most convenient way for any particular application. 

In addition the mathematical properties of the Duhem-Margules equation are dealt with by Kamke. ... 

Sb. Vapor Pressure Curves for Nonideal Systems.—The general nature of the vapor pressure cuiwes showing positive and negative deviations are depicted in Fig. 22, A and B, respectively these results refer to a constant temperature. At any given composition, the slopes of the two partial vapor pressure curves are related by the Duhem-Margules equation. Thus, if the... 

Sc. Liquid and Vapor Compositions.—Some general rules concerning the relative compositions of liquid and vapor in equilibrium, which are applicable to systems of all types, may be derived from the Duhem-Margules equation, using the form of (35.1). Since the increase in the mole fraction of one component of a binary mixture must be equal to the decrease for the other component, dNi is equal to — c/N2, as seen in 34b hence equation (35.1) may be written as... 

The relationships between the constants ai, /Si and otz, fiz, which are different from those in the Margules equations (35.7) and (35.8), can be derived by means of the Duhem-Margules equation in a manner similar to that described above. It is then found that... 

In this section we consider the equilibrium between a condensed phase and vapor. We derive the Duhem-Margules equation and investigate its application to the determination of vapor fugacities. 

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