The Duhem-Margules equation

When we divide both sides of Eq. 12.8.6 by dxA, we obtain the Duhem-Margules equation  

If we assume the gas mixture is ideal, the fugacities are the same as the partial pressures, and the Duhem-Margules equation then becomes 

Pb Xa (ijjnary liquid mixture equilibrated with ideal gas, constant T and p) 

To a good approximation, by assuming an ideal gas mixture and neglecting the effect of total pressure on fugacity, we can apply Eq. 12.8.20 to a liquid-gas system in which the total pressure is not constant, but instead is the sum of pa and pb- Under these conditions, we obtain the following expression for the rate at which the total pressure changes with the liquid composition at constant T  

We can use Eq. 12.8.21 to make several predictions for a binary liquid-gas system at constant T. 

---

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. 

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.)... 

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. 

The Duhem-Margules equation is usually written in either isothermal or isobaric forms. It is most frequently used in the determination of the fugacities of vapor in equilibrium with liquid. At constant temperature Eq. (9-141) reduces to... 

The Duhem-Margules equation [Eq. (9-141)] at constant pressure becomes... 

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. 

Another way to obtain integral thermodynamic functions is provided by the Duhem-Margules equation. The partial molar function AZ is plotted versus y (the composition of the alloy is represented by the formula A B) and then is integrated from... 

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.

There are two limiting solutions to the Duhem Margules equation ... 

Departures from the above equations due to deviations from the law of the perfect gas mixture do not usually exceed a small percentage except at pressures above atmospheric or if there is association in the vapour phase, such as occurs in the case of formic and acetic acids. A procedure for applying the Duhem-Margules equation allowing for deviations from the gas law has been described by Scatchard and Raymond. ... 

It will be found best to plot the logarithms of the partial pressures. The data do not agree at all well with the Duhem-Margules equation, and it appears therefore that there is considerable experimental error. 

---