Calculation of phase equilibria from the chemical potentials

28 Calculation of phase equilibria from the chemical potentials 

In the foregoing we have discussed the determination of the chemical potentials as functions of the temperature, pressure, and composition by means of experimental studies of phase equilibria. The converse problem of determining the phase equilibria from a knowledge of the chemical potentials is of some importance. For any given phase equilibrium the required equations are the same as those developed for the integral method. The solution of the equation or equations requires that a sufficient number of 

The equations are transcendental equations, which must be solved by approximation methods. This presents no problem with the use of modem computers. However, it is still appropriate to discuss graphical aids to the solution of the equations. We discuss only one example. Let us consider the equilibrium between a solid solution and a liquid solution at a constant pressure. For the present we choose the pure solid phases and the pure liquid phases as the standard states of the two components for each phase. The equation for equilibrium for the first component is 

The equation for the second component is the same with change of subscripts. The problem is to determine values of xt and zl at specified temperatures by the use of the two equations. We define the quantity A [T, P] as equal to both sides of Equation (10.198). This quantity is the change of the chemical potential on mixing for the liquid phase that has been defined previously however, for the solid phase the standard state is now the pure liquid component. A similar definition is made for Ap%[T, P], The conditions of equilibrium then become 

We must emphasize that these conditions are valid only when the standard state of a component is the same for both phases. We now choose a temperature, calculate values of Aju and Aju for each phase for various mole fractions and plot Aju against Aju for each phase. Two curves are obtained, one for each phase. The point of intersection of the curves gives the equilibrium values of Aju and Aju according to Equations (10.199) and (10.200). The calculation of x1 by the use of the left-hand side of Equation (10.198) and zl by the use of the right-hand side of Equation (10.198) is then relatively simple. If no intersection occurs, then no equilibrium between the two phases exists at the chosen temperature and pressure. The calculation must be repeated for each temperature. Activities could be used in place of Apu provided the same standard state for a component is used for both phases. 

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