Gibbs-Duhem equation three-component systems

A one-component system may have a maximum of three phases in equilibrium. Therefore, the three possible Gibbs-Duhem equations are... 

In the previous examples we have assumed that all comonents are present in all of the phases, and we have not introduced any chemical reactions or restrictions. When a component does not exist in a phase, the mole number of that component is zero in that phase and its chemical potential does not appear in the corresponding Gibbs-Duhem equation. As an example, consider a two-component, three-phase system in which two of the phases are pure and the third is a solution. The Gibbs-Duhem equations are then... 

First we consider the binary systems when no inert gas is used. When only one of the components is volatile, the intensive variables of the system are the temperature, the pressure, and the mole fraction of one of the components in the liquid phase. When the temperature has been chosen, the pressure must be determined as a function of the mole fraction. When both components are volatile, the mole fraction of one of the components in the gas phase is an additional variable. At constant temperature the relation between two of the three variables Pu x1 and yt must be determined experimentally the values of the third variable might then be calculated by use of the Gibbs-Duhem equations. The particular equations for this case are... 

The experimental studies of three-component systems based on phase equilibria follow the same principles and methods discussed for two-component systems. The integral form of the equations remains the same. The added complexity is the additional composition variable the excess chemical potentials become functions of two composition variables, rather than one. Because of the similarity, only those topics that are pertinent to ternary systems are discussed in this section of the chapter. We introduce pseudobinary systems, discuss methods of determining the excess chemical potentials of two of the components from the experimental determination of the excess chemical potential of the third component, apply the set of Gibbs-Duhem equations to only one type of phase equilibria in order to illustrate additional problems that occur in the use of these equations, and finally discuss one additional type of phase equilibria. 

This is the Gibbs-Duhem equation for a two-dimensional system with volume and pressure being replaced by area and spreading pressure, respectively. For three dimensional systems where all components experience the same total pressure at equilibrium, all components in the two dimensional systems will experience the same spreading pressure. 

For a three-component system, the Gibbs-Duhem equation and the condition for ng volume flow are as follows (equations 16.4.5 and 16.4.6) ... 

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