first- and second-order transitions
Not all phase changes feel the same. Boiling water is abrupt and dramatic — bubbles, a great gulp of latent heat, a sudden jump in volume. But a magnet quietly losing its magnetism as it warms, or liquid helium turning superfluid, happens smoothly, with no sudden lurch. Sorting transitions by how abruptly they happen is what 'order' of a transition means.
A first-order transition is the abrupt kind. It absorbs or releases latent heat at a fixed temperature, and properties like volume, density, and energy jump discontinuously. Melting, boiling, and sublimation are all first-order — that is why two phases coexist during them and the temperature pauses. A second-order (or continuous) transition has no latent heat and no jump in those basic properties; instead, the change shows up in how those properties respond — heat capacity, for instance, can spike or kink right at the transition.
The distinction, sharpened by Paul Ehrenfest and later refined, matters because the two kinds behave very differently. First-order transitions can be 'overshot' — water can supercool below freezing or superheat above boiling before suddenly flipping. Second-order transitions cannot be overshot and are the home of critical phenomena, where fluctuations grow without limit and many different materials suddenly obey the same universal laws.
Boiling water is first-order: it soaks up latent heat and its volume leaps a thousandfold. A piece of iron losing its magnetism at 770 °C is second-order: it gives off no latent heat and nothing jumps — only its magnetic response melts away smoothly.
First-order jumps with latent heat; second-order changes smoothly.
The classification by 'order' traces to Ehrenfest, who keyed it to which derivative of the free energy first becomes discontinuous. The modern view treats the line as first-order versus continuous; the critical point at the end of the liquid–gas line is itself a second-order, continuous transition.