Clapeyron equations, Gibbs-Duhem

The common characteristics of phase transitions are that the Gibbs energy is continuous. Although the conditions of equilibrium and the continuity of the Gibbs energy demand that the chemical potential must be the same in the two phases at a transition point, the molar entropies and the molar volumes are not. If, then, we have two such phases in equilibrium, we have a set of two Gibbs-Duhem equations, the solution of which gives the Clapeyron equation (Eq. (5.73))... 

An alternative proof of the Clausius-Clapeyron Equation emerges (Frame 50) using the Gibbs Duhem Equation. 

The Clapeyron equation relates pressure to temperature, and hence boiling or melting points can be calculated with changing pressure. By using Eq. (1.156), we can equate the Gibbs-Duhem equation for two phases... 

Once a state point of coexistence is established, additional state points can be determined expeditiously through application of the Gibbs-Duhem integration method [48,85,86]. In this approach a differential equation for the coexistence line is used to guide the establishment of state points away from the known coexistence point. The most well known such formula is the Clapeyron equation [41]... 

Equation (5.2.4) is called the Gibbs-Duhem equation. It shows that changes in the intensive variables T, p and cannot all be independent. We shall see in Chapter 7 that the Gibbs-Duhem equation can be used to understand the equilibrium between phases, and the variation of boiling point with pressure as described by the Clausius-Clapeyron equation. 

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