Drop an ice cube into a warm glass and watch it: for a while it just sits there, shrinking, while the water around it stays stubbornly at 0\,^\circ\text{C}. Then the last sliver vanishes and the temperature climbs again. Push a kettle further and, at 100\,^\circ\text{C}, the liquid erupts into vapour. Heat a bar magnet past a certain temperature — its Curie point — and its magnetism simply switches off. Cool certain metals below a few kelvin and their electrical resistance drops abruptly to exactly zero: they become superconductors. In every one of these cases a substance abruptly changes character as you turn a single knob — temperature, or pressure, or magnetic field.

These are phase transitions, and the astonishing thing is not that they happen but that they are all, deep down, the same kind of event. This page builds the one idea that ties them together: a phase transition is a place where two competing arrangements of matter trade places, and the way they trade — smoothly or with a jolt — sorts every transition in nature into just two families, first order and second order. Get that distinction straight and melting ice, a boiling kettle, a demagnetising magnet and a switching superconductor all fall into one picture.

Free energy decides who wins

Why should a substance ever change phase at all? Because at fixed temperature T and pressure p, nature is a relentless minimiser: it settles into whichever arrangement has the lowest Gibbs free energy G. Ice and liquid water are two rival phases, each with its own G(T,p). At low temperature the rigid crystal wins; at high temperature the disordered liquid wins. The transition happens at exactly the temperature where the two curves cross — where the phases have equal Gibbs free energy:

On one side of that line the solid's G is lower and the substance is solid; on the other side the liquid's G is lower and it melts. The whole zoo of transitions is really this one contest — competing phases with competing free energies, swapping the lead as you change the conditions. Everything else on this page is just a careful look at how G behaves as the phases swap.

The map of phases: the p–T diagram

Plot pressure against temperature and every phase claims a territory. The (T,p) plane splits into regions — solid, liquid, gas — each being the phase whose G is lowest there. The borders between territories are the coexistence lines where two phases tie. Three of them meet at one special spot, the triple point, where solid, liquid and gas coexist simultaneously (for water, at 0.01\,^\circ\text{C} and a mere 611\ \text{Pa}).

Follow the liquid–gas line up and to the right and something strange happens: it does not run forever. It stops, at the critical point. Beyond it the distinction between liquid and gas simply dissolves — there is one continuous supercritical fluid, dense as a liquid yet space-filling as a gas, with no surface between the two. Reveal the diagram step by step to see the regions, the triple point and the critical point appear.

Two families: first order and second order

Here is the heart of the page. Since G is continuous across every transition, we classify transitions by how its derivatives behave. The two most important first derivatives of the Gibbs free energy are the entropy and the volume:

A first-order transition is one where these first derivatives jump. The entropy S changes discontinuously by \Delta S, and so a chunk of heat — the latent heat L = T\,\Delta S — must be absorbed or released at constant temperature to get across. The volume V jumps too (ice is less dense than water; steam is far less dense than either). Because the two phases genuinely differ at the boundary, they coexist there — ice and water sit together in the glass at 0\,^\circ\text{C}. Geometrically, G(T) is continuous but has a kink: its slope changes abruptly. Melting, boiling and sublimation are all first order.

A second-order (continuous) transition is gentler. Now the first derivatives S and V are continuous — there is no latent heat and no sudden density jump. Instead it is the second derivatives that misbehave: quantities like the heat capacity

the compressibility and the magnetic susceptibility diverge or jump. The ferromagnetic Curie point, the superfluid lambda transition (whose heat-capacity curve is shaped like the Greek letter \lambda), superconductivity, and the liquid–gas critical point itself are all second order.

Latent heat and the temperature plateau

The signature of a first-order transition is a flat step on a heating curve. Add heat to a block of ice at a steady rate and its temperature climbs — until it reaches 0\,^\circ\text{C}. There it stops climbing. Keep pouring in heat and the temperature refuses to budge, because every joule is going into breaking the crystal bonds and rearranging molecules, not into speeding them up. Only once the last of the ice has melted does the temperature start rising again. The same plateau appears at 100\,^\circ\text{C} while the water boils — a far longer plateau, because vaporising costs much more than melting.

For water, per kilogram:

• Latent heat of fusion (melting): L_\text{fus} \approx 334\ \text{kJ/kg};
• Latent heat of vaporisation (boiling): L_\text{vap} \approx 2260\ \text{kJ/kg}.

The heating curve below plots temperature against total heat added for one kilogram of water, starting from ice at -20\,^\circ\text{C}. The two horizontal plateaus are exactly the latent heats being paid; the boiling plateau is almost seven times as wide as the melting one.

Worked example — melting then boiling

How much heat does it take to turn 0.5\ \text{kg} of ice already at 0\,^\circ\text{C} into steam at 100\,^\circ\text{C}? Add the three contributions:

• Melt the ice: Q_1 = m\,L_\text{fus} = 0.5 \times 334 = 167\ \text{kJ};
• Warm the water 0 \to 100\,^\circ\text{C}: Q_2 = m\,c\,\Delta T = 0.5 \times 4.18 \times 100 = 209\ \text{kJ};
• Boil the water: Q_3 = m\,L_\text{vap} = 0.5 \times 2260 = 1130\ \text{kJ}.

Notice how the two latent heats (the flat steps) dwarf the warming in between — and how boiling alone costs more than everything else put together.

The order parameter and critical points

A second-order transition needs a subtler description than "a jump", because nothing jumps. The right language is the order parameter: a quantity that is nonzero in the ordered phase and zero in the disordered one. For a ferromagnet it is the magnetisation M — the iron is spontaneously magnetised below the Curie temperature T_c and unmagnetised above it. For the liquid–gas critical point it is the density difference \rho_\text{liq} - \rho_\text{gas}, which distinguishes the two fluids below the critical point and collapses to zero at it.

The defining feature of a continuous transition is that the order parameter does not jump to zero — it slides continuously to zero as T \to T_c from below, following a power law with a critical exponent \beta:

The remarkable discovery of the twentieth century is universality: wildly different systems — a magnet, a fluid at its critical point, an alloy ordering — share the same value of \beta. It depends only on gross features like the dimensionality and the symmetry, not on the microscopic chemistry. The graph below shows M(T) falling to zero at T_c; drag the slider to change \beta and watch how the curve approaches zero — steeply for small \beta (a nearly vertical drop, as for the 3-D Ising magnet with \beta \approx 0.33), gently for the mean-field value \beta = 0.5.

Right at the critical point a second dramatic thing happens: fluctuations grow to enormous size. The correlation length — the distance over which molecules move in concert — diverges, and density ripples swell until they scatter all wavelengths of light at once. A clear fluid turns milky and opaque: critical opalescence, a phenomenon you can see with your own eyes as matter teeters between two phases.

Sliding along a first-order line: Clausius–Clapeyron

The coexistence lines on the phase diagram are not arbitrary curves — their slopes are fixed by the very latent heat and volume change that define a first-order transition. Setting G_1 = G_2 all along the boundary and differentiating gives the Clausius–Clapeyron relation:

This one formula explains a famous quirk. For almost every substance the melting line tilts to the right (higher pressure raises the melting point). But water is a rebel: ice is less dense than liquid water, so \Delta V is negative going from solid to liquid, and the melting line tilts left — pressure lowers water's melting point. Squeeze ice and it melts. That backwards slope, straight out of Clausius–Clapeyron, is part of why glaciers flow and why an ice skater glides on a thin film of meltwater.