'Chemical' First-order Phase Change First order phase changes occur when one local minima of the Gibbs Free Energy becomes deeper than another. Thus at 1 atm and 99 °C, the Gibbs Free Energy of liquid water is less than the Gibbs Free Energy of steam at those conditions (though both states form local minima). As the temperature increases, the relative depths of the two minima change: at 100 °C they are the same depth, and above 100 °C the vapour-phase minimum is lower.

I'm wondering if similar first-order changes can occur in chemically reacting systems? It seems to me that equilibrium constants always change continuously with temperature: are there any situations in which this isn't the case, and there exist discrete jumps in chemical compositions? And if these jumps in the deepest minima of the Gibbs Free Energy never occur in chemically reacting systems, do we know why not? Of course, situations in which a chemical reaction is accompanied by a phase change don't count!
Well, let's see. As you know, the equilibrium constant is determined by the reaction's $\Delta_rG:\;K=e^{-\frac{\Delta G}{RT}}$, so a discrete jump in the former means a discrete jump in the latter. What is that $\Delta_rG$? That's the sum of $\Delta_fG$ of products minus that of starting compounds. Now what is the $\Delta G$ of formation of a compound? A continuous function, unless there is a phase transition, which by your problem definition is not the case. Well, a linear combination of a few continuous functions is still a continuous function.