Some Implications of Mathematics in Thermodynamics

The mathematics of thermodynamics places restrictions on how certain thermodynamic curves must appear when they are graphed.

There is one curve that is often drawn incorrectly by beginners and that is how $G$ changes with temperature at constant pressure and composition. Crusty old materials scientists are fond of pointing out, when this particular mistake is made,¹ that knowing how to take derivatives is important.

An example of the plot is given below:

Figure 23-1: How should $G$ behave when only $T$ is changing?

First, what is the slope of this curve?

$$\input{equations/16-1A}$$ (23-1)

Second, what is the curvature of the curve?

$$\input{equations/16-2A}$$ (23-2)

Therefore the curve must appear as follows:

Please learn these rules for drawing curves for the future. Someone is likely to ask you how they are drawn--if you make this particular mistake, please say that someone else besides me taught you thermo...

Question: What if another phase is considered? How would it be drawn? What conclusions can be made from the graph?