# Common Tangent Construction of Phase Diagrams

The free energy of mixing of a system, Gmix, depends on many conditions such as temperature, pressure, and composition. By fixing all variables except temperature and composition, one can generate a family of Gmix vs. composition curves:

(figure 1 is an example of a family of Gmix vs. composition curves at different temperatures. See DeHoff fig 10.1)

For a given temperature, each phase will have its own Gmix vs. composition curve. At that particular temperature the system will seek the lowest free energy; which is to say it will become the phase with the most negative Gmix. For example suppose there existed a system which could exist as either a liquid or a solid phase called alpha:

(Figure 2 is DeHoff figure 10.10d, which shows Gmix vs. composition for two separate phases. The two lines do not cross)

The above figure suggests that for this particular temperature the system would rather be in the alpha phase no matter what the composition is.

Let's assume we took the previous liquid/alpha example to a temperature high enough to melt the entire sample, then cooled it. In the melted state, the Gmix line of the liquid will be below that of the liquid (the opposite of figure 2). As we decrease the temperature, there may be compositions where Gmix curve of the alpha phase is more negative than the liquid phase. When this is so, it is possible to draw a line that is tangent to both curves:

(Figure 3 is a graph like the previous two, except the two curves do cross and are connected by a tangent as described above)

For any composition between the two tangent points, the Gmix will now lie on the tangent line between these points. It is within this range of compositions that the two phases will exist in equilibrium. As temperature continues to decline, the compositions marking the two-phase region will change. By plotting these compositions at the temperature where they occur, a phase diagram can be generated:

(Figure 4 is an MPEG of five separate images (see DeHoff 10.9a-e) which show the generation of a phase diagram. I will include instructions to click through them one image at a time rather than run it full-time, otherwise it'd be impossible to watch)

Under certain conditions a miscibility gap may form. A miscibility gap occurs due to a rise in a phase's Gmix vs. composition curve. The result is a region where two phases will form from a previously one-phase system:

(Figure 5 is also an MPEG of the same format as figure 4 showing how the Gmix curve causes the miscibility gap to form. See DeHoff fig. 10.11)

The final paragraph will discuss how to form a binary eutectic diagram by superimposing two ideal solution graphs with a miscibility graph:

(The final figure will be Dehoff fig. 10.16d showing a binary eutectic diagram and the three sets of curves used to make it.)

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