Duhem-Margules relation

The surface tension of mixtures of water and ethyl alcohol at 25 C has been measured by Bircumshaw (J. Chem. Soc. 1922, 887). Partial vapour pressures of these laiixtures also at 25 C have been determined by Dobson (J. Chem. Soc. 1925, 2866). These values, smoothed in accordance with the requirements of the Duhem-Margules relation are given with related quantities at rounded mole fractions in table 1 (Guggenheim and Adam, Proc. Roy. Soc. A 1933, 139, 218). The subscript 1 relates to water and 2 to alcohol, x denotes mole fraction, p partial pressure, and y surface tension. 

We may use either the equation between y and or that between y and pi. We shall use the former. Their equivalence is an immediate consequence of the Duhem-Margules relation. 

The functions G, iif, S, V are directly proportional to the total amount of the mixture. If all numbers Ni are doubled, these functions also become twice as large. It follows that in a mixture of two components, o and r, the chang e of go with concentration X is connected with the change in gr according to the well-known Gibbs-Duhem (often also called Duhem-Margules) relation ... 

Losungen der Gleichung von Duhem-Margules Solutions of the Duhem-Margules relation... 

If the activities and activity coefficients are substituted for partial vapour pressures, they must obey the Duhem-Margules relation. If this is the case, then they are called thermodynamically consistent. Therefore with this equation there can be checked whether experimental or otherwise derived activities or activity coefficients are thermodynamically consistent in their dependence on composition. The same holds true for the fugacities discussed later. 

To be able to apply the Duhem-Margules relation exactly to a variable pressure at constant temperature, the effect of the pressure must be expressed by an additional term. From the relation proposed by Gibbs and Duhem... 

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... 

For mixtures containing a liquid and a vapour phase the most important consequence of the equilibrium conditions is the Duhem-Margules equation. If we treat the vapour as a mixture of ideal gases, then the Duhem-Margules equation gives a relation between the partial vapour pressures pjf, p, . ..,pj. .. of the components of the solution in the form... 

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. 

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

Mathematical Investi-g ations.—The whole question of the relations between the composition of liquid mixtures and (a) the partial pressures of the vapours of the components (6) the composition of the vapour, has been discussed mathematically by Duhem (12), Margules (13),... 

---