Limited uses of the Gibbs-Duhem equation

Three different uses of the Gibbs-Duhem equation associated with the integral method are discussed in this section (A) the calculation of the excess chemical potential of one component when that of the other component is known (B) the determination of the minimum number of intensive variables that must be measured in a study of isothermal vapor-liquid equilibria and the calculation of the values of other variables and (C) the study of the thermodynamic consistency of the data when the data are redundant. 

Gibbs energy. Let us assume that A/iE[T, P, x] has been determined over a range of composition at constant temperature and pressure. Then, by the Gibbs-Duhem equation for the single phase at constant temperature and pressure, 

An alternative calculation, particularly applicable when the experimental studies can be carried out over the entire range of composition, involves the direct calculation of the molar excess Gibbs energy. We have 

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 

When d/i2 is eliminated from these two equations, Equation (10.29) is used to express as a function of P and yt with the relation 

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