Using the Idea of Tangent Intercepts

If we then divide out the fractions on the right, thus changing the numbers of mols back to mol fractions, we find 

A little algebra (Problem 6.7) shows that Eq. 6.L is identical to the solution to Example 6.4.  

Example 6.5 is clearly a longer and messier way to do by tangent slopes what we did more easily by tangent intercepts in Example 6.4. So why bother Because some functions are easier to do by tangent slopes, and this method appears in the historical literature, so you must understand it to understand that literature. Most often we use tangent intercepts and Eqs. 6.4 and 6.5 

Example 6.6 For liquid mixtures of a and /3 (two imaginary chemicals) at some fixed T and P the Gibbs energy per mol of the mixture is given by 

observe that no real mixtures have this simple an equation for Gibbs energy as a function of composition however, this equation leads to simple mathematics, so please bear with it. Here we know that we have two relations to satisfy, namely that the partial molar Gibbs energy of a is the same in both phases and that the partial molar Gibbs energy of /3 is the same in both phases. 

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