If we add a small amount of heat δq to a liquid at its boiling point Tb, it could either increase its temperature a small amount dT = δq/Cv, or suffer a phase change into vapor. Experimentally, we see that liquids remain at Tb until the phase change is complete, and only then continue increasing their temperature. Why is this? I want to prove this macroscopically, that is, from thermodynamics, but I don't know where to start. Any leads?