Why do disproportionation reactions take place?
"Stability" is not really a driving force of a reaction. That would be the change in Gibbs Free Energy or ∆G. That takes into account both entropy and enthalpy, as well as concentrations of reactants and products, and temperature.

If you haven't yet discussed these in class, you should be able to read about these concepts by looking up "Gibbs Free Energy" in your textbook's index.