Several ongoing theoretical and experimental projects are exploring the possibility of a QCD critical point and phase transition. Here, we propose to study the production of entropy as a signature for a first-order chiral phase transition. Based on the linear sigma model, we couple the order parameter sigma to a quark fluid to describe the nonequilibrium cooling of the fireball in a heavy-ion collision. We estimate the relative increase in entropy per baryon number S/N to reach its largest value of up to 130% for a first-order phase transition at low beam energies. The lifetime of the quark-gluon plasma (QGP) at different types of transition is also investigated.