Phase separation of a liquid mixture

A binary liquid mixture in a system maintained at constant T and p can spontaneously separate into two liquid layers if any part of the curve of a plot of AGm(niix) versus xa is concave downward. To understand this phenomenon, consider Fig. 11.4. This figure is a plot of AGm(mix) versus xa- It has the form needed to evaluate the quantities (/ta - A a) and (/tb - / b) the variant of the method of intercepts described on page 233. On this 

T/jermoafynam/csandC/jem/stry, second edition, version 3 2011 by Howard DeVoe. Latest version www.chem.umd.edu/theniiobook 

In order for two binary liquid phases to be in transfer equilibrium, fiA must be the same in both phases and /tb must also be the same in both phases. The dashed line in the figure is a conunon tangent to the curve at the points labeled a and p. These two points are the only ones having a conunon tangent, and what makes the common tangent possible is the downward concavity (negative curvature) of a portion of the curve between these points. Because the tangents at these points have the same intercepts, phases a and P of 

Now consider point 1 on the curve. A phase of this composition is unstable. It will spontaneously separate into the two phases of compositions x and x, because the Gibbs energy per total amount then decreases to the extent indicated by the vertical arrow from point 1 to point 2. We know that a process in which G decreases at constant T and in a closed system, with expansion work only, is a spontaneous process (Sec. 5.8). 

To show that the arrow in Fig. 11.4 represents the change in G/n for phase separation, we let y represent the vertical ordinate and write the equation of the dashed line through points a and P (y as a function of xa)  

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