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First-order transitions (e.g., liquid-gas) involve a discontinuous jump in density and latent heat; Gibbs free energy is continuous but its first derivatives (entropy, volume) are discontinuous. Second-order transitions (e.g., ferromagnetic) show no latent heat or density jump; Gibbs energy and its first derivative are continuous, but second derivatives diverge.
Phase transitions are among the most striking phenomena in nature: a substance abruptly changes character — liquid to gas, paramagnet to ferromagnet, normal metal to superconductor — at a precise temperature and pressure. The Ehrenfest classification organizes this diversity into two fundamental categories based on which derivatives of the Gibbs free energy G(T, P) are discontinuous at the transition point.
For a first-order transition, G itself is continuous across the transition (if it weren't, the system wouldn't choose that point), but its first partial derivatives are discontinuous. Since S = −(∂G/∂T)_P and V = (∂G/∂P)_T, a discontinuous first derivative means a jump in entropy — which is the latent heat L = TΔS — and a jump in volume (density change). This is the familiar liquid-gas transition: when water boils at 100°C, it absorbs 2260 J/g of latent heat while its density drops by a factor of ~1600. The two phases coexist along the transition curve, with equal Gibbs free energies. Moving along the coexistence curve toward the critical point, the latent heat and density jump shrink continuously, reaching zero at the critical point — where the transition changes character entirely.
A second-order transition (also called a continuous transition) has no latent heat and no coexistence: the system transforms smoothly in the sense that the order parameter — the quantity that characterizes the ordered phase — grows continuously from zero rather than jumping. For a ferromagnet, the order parameter is spontaneous magnetization, which appears below the Curie temperature and grows continuously. However, the second derivatives of G diverge at the transition: the heat capacity C_P = −T(∂²G/∂T²)_P and the compressibility κ_T = −(1/V)(∂²G/∂P²)_T both blow up. This divergence reflects the growth of fluctuations: near a second-order transition, fluctuations occur at all length scales simultaneously, making the system scale-invariant and the correlation length infinite.
The free energy framework unifies both cases with a single geometric picture. At a first-order transition, the G(T) curves for two phases cross: each phase has lower G in its own stability region, and they are equal exactly on the coexistence line. At a second-order transition, G approaches the same value from both phases continuously, with matching first derivatives. The practical diagnostic is straightforward: if heating a substance causes it to absorb latent heat at a fixed temperature (with coexisting phases), the transition is first-order. If instead the heat capacity diverges sharply without a finite latent heat, it is second-order. Both types are driven by the competition between energy, which favors ordered states, and entropy, which favors disorder — encoded together in the free energy G = H − TS.