Phases, Coexistence, and the Classification of Transitions

A single substance held at fixed particle number is described by two intensive variables, say temperature and pressure. Over most of that plane the equilibrium state is a smooth function of $(T, P)$ : heat it a little and the density, entropy, and energy shift a little. Along certain curves this smoothness breaks. Crossing the boiling line at fixed pressure, the density drops by three orders of magnitude across an interval of zero width in temperature; the entropy jumps; the system absorbs a finite quantity of heat while its temperature does not move. A phase transition is a locus in the space of control variables where the free energy per particle fails to be analytic, and the different phases are the regions of analyticity it separates.

The thermodynamics of Module 01 assigns each equilibrium state a Gibbs free energy $G (T, P, N) = N g (T, P)$ , with $g$ the free energy per particle. Every equilibrium property follows from derivatives of $g$ : the entropy per particle is $s = - (\partial g / \partial T)_{P}$ , the volume per particle is $v = (\partial g / \partial P)_{T}$ . Where $g$ is smooth, so are $s$ and $v$ . A phase transition is a point where some derivative of $g$ is discontinuous or divergent, and the whole classification of transitions is a classification of which derivative first goes bad.

Phases and the phase diagram

A phase is a spatially uniform equilibrium state with its own equation of state. Solid, liquid, and gas are the three phases of a simple substance; distinct crystal structures, magnetically ordered and disordered states, and normal and superfluid liquids are further examples. Two phases can occupy the same container only along the boundaries where their free energies per particle coincide.

The equation $g_{1} (T, P) = g_{2} (T, P)$ is one relation between two variables, so its solutions form a curve in the $(T, P)$ plane: the coexistence curve of the two phases. Off the curve, whichever phase has the lower $g$ is the stable one; the system minimizes its Gibbs free energy at fixed $(T, P)$ . The plane divided into regions of lowest- $g$ phases, with the coexistence curves as their borders, is the phase diagram.

The

T

–

P

phase diagram of a simple substance; the sublimation, fusion, and vaporization curves meet at the triple point, and the vaporization curve terminates at the critical point beyond which liquid and gas are one phase.

Three coexistence curves meet at the triple point, where solid, liquid, and gas are simultaneously in equilibrium; its temperature and pressure are fixed numbers for each substance. The vaporization curve does not continue forever. It ends at the critical point $(T_{c}, P_{c})$ , beyond which no discontinuity separates liquid from gas: a path that loops around the critical point carries the dense liquid continuously into the dilute gas without ever crossing a phase boundary. The fusion curve, by contrast, has no known terminus, because solid and liquid differ by a symmetry (the crystal breaks continuous translation) that a state either has or lacks.

The number of intensive variables that can be tuned independently while a set of phases remains in coexistence is fixed by the Gibbs phase rule. For a system of $C$ chemical components and $Φ$ coexisting phases,

f = C - Φ + 2,

with $f$ the number of degrees of freedom. A single phase of a pure substance ( $C = 1, Φ = 1$ ) has $f = 2$ : temperature and pressure vary freely over an area. Two coexisting phases have $f = 1$ , a curve. Three phases have $f = 0$ , an isolated point — the triple point.¹

The Clausius-Clapeyron relation

The slope of a coexistence curve is fixed by the difference in entropy and volume between the two phases. Along the curve $g_{1} (T, P) = g_{2} (T, P)$ at every point, so moving an infinitesimal step along it keeps the equality:

d g_{1} = d g_{2} .

Each phase obeys the Gibbs-Duhem relation $d g = - s d T + v d P$ , with $s$ and $v$ the entropy and volume per particle. Substituting and collecting terms,

- s_{1} d T + v_{1} d P = - s_{2} d T + v_{2} d P ⟹ \frac{d P}{d T} = \frac{s _{2} - s _{1}}{v _{2} - v _{1}} .

The entropy difference is measured by the latent heat $L = T Δ s$ , the heat absorbed per particle in converting phase $1$ to phase $2$ at fixed temperature. Writing $Δ v = v_{2} - v_{1}$ ,

The relation is exact, not a model. It ties three measurable quantities together: measure any two of slope, latent heat, and volume change, and the third follows.

The Clausius-Clapeyron slope reads off the coexistence curve; at each point

d P / d T

equals

L / (T Δ v)

, the latent heat over temperature times the volume jump between phases.

For vaporization the volume change is dominated by the gas, $Δ v \approx v_{gas} \approx k_{B} T / P$ if the vapor is treated as ideal, and the latent heat is nearly constant over a modest range. Substituting turns the relation into $d P / d T \approx L P / (k_{B} T^{2})$ , which integrates to the approximate vapor- pressure law $P (T) \propto exp (- L / k_{B} T)$ .

First-order transitions

The transitions crossed along the coexistence curves share a signature: a finite latent heat and a finite volume change. Both are first derivatives of the Gibbs free energy, $s = - (\partial g / \partial T)_{P}$ and $v = (\partial g / \partial P)_{T}$ , and both jump discontinuously across the curve. A transition in which a first derivative of $g$ is discontinuous is a first-order transition.

The discontinuity in the first derivative means $g$ itself has a kink: the two phases correspond to two branches $g_{1} (T, P)$ and $g_{2} (T, P)$ , each analytic on its own, and the physical free energy is their lower envelope $g = min (g_{1}, g_{2})$ . Where the branches cross, the slope changes abruptly, and the slope is the entropy. The latent heat $L = T (s_{2} - s_{1})$ is the size of that jump.

At a first-order transition the physical free energy is the lower envelope of two analytic branches; its slope (the entropy) jumps at the crossing, and the entropy versus temperature shows a step of height

L / T

Because the two phases have genuinely different free energies away from the transition, a first-order transition supports metastability. The dotted continuations in the figure are the analytic branches carried past the crossing: a superheated liquid or supercooled vapor sits on the higher branch, locally stable against small fluctuations but globally unstable, and it decays to the lower branch by nucleation once a large enough droplet or bubble forms. Latent heat, metastability, hysteresis, and phase coexistence with a sharp interface are the interlocking hallmarks of first-order behavior.

Continuous transitions and the order parameter

At the critical point the vaporization curve ends, and the distinction between liquid and gas disappears. Approaching it along the coexistence curve, the density difference between the two phases shrinks to zero and the latent heat vanishes. The transition that occurs exactly at the critical point has no latent heat and no volume jump: the first derivatives of $g$ are continuous. What diverges instead is a second derivative — the isothermal compressibility $κ_{T} = - v^{- 1} (\partial v / \partial P)_{T}$ grows without bound at the critical point. A transition in which the first derivatives of $g$ are continuous but a higher derivative diverges or jumps is a continuous transition (older usage: second-order, or critical).

Continuous transitions are organized by an order parameter: a quantity that is zero in the symmetric (disordered) phase and nonzero in the ordered phase, turning on continuously as the transition is crossed.

Liquid-gas critical point — the order parameter is the density difference $ρ_{liq} - ρ_{gas}$ , which falls to zero at $T_{c}$ .
Ferromagnet — the order parameter is the spontaneous magnetization $m$ , zero above the Curie temperature and nonzero below it, with no external field.
Superfluid and superconductor — the order parameter is a complex amplitude whose magnitude sets the condensate or pair density.

Across a continuous transition the order parameter rises from zero as the temperature drops through

T_{c}

, vanishing with a characteristic power law; above

T_{c}

it is identically zero.

The value of the order parameter is not fixed by the control variables alone. In the ferromagnet below $T_{c}$ the magnetization can point up or down with equal free energy; the system selects one, breaking the up-down symmetry of its Hamiltonian. This spontaneous symmetry breaking is the defining feature of the ordered phase, developed in the mean-field lesson.

The Ehrenfest classification and its limits

Ehrenfest proposed grading transitions by the lowest derivative of $g$ that is discontinuous: first-order if a first derivative jumps, second-order if the first derivatives are continuous but a second derivative jumps, and so on. The scheme is clean but wrong in detail for the continuous case. At real critical points the second derivatives — heat capacity, compressibility, susceptibility — do not jump by a finite amount; they diverge. The Ehrenfest picture of a finite step assumes each derivative stays finite, which fails whenever fluctuations grow large, precisely the regime near a critical point.

The modern classification keeps only the top-level split:

First-order — a first derivative of the free energy is discontinuous; there is a latent heat, a jump in the order parameter, phase coexistence, and metastability.
Continuous — the first derivatives are continuous, the order parameter vanishes continuously, and higher derivatives (response functions) typically diverge with power-law singularities. There is no latent heat and no coexistence at the transition itself.

The divergences at a continuous transition are governed by critical exponents that prove to be shared across wildly different systems, a universality quantified in the later lessons. The singularity in $g$ that produces them is the central object: the finite-size partition function is a finite sum of exponentials, hence perfectly analytic, so the non-analyticity of a genuine phase transition emerges only in the thermodynamic limit $N \to \infty$ . A phase transition is a property of the infinite system, approached but never reached by any finite one.

Feature	First-order	Continuous
Lowest singular derivative of $g$	first ( $s$ , $v$ )	second ( $C$ , $κ$ , $χ$ )
Latent heat	finite	zero
Order parameter at $T_{c}$	jumps	vanishes continuously
Coexistence / interface	yes	no
Metastability, hysteresis	yes	no
Response functions	finite	diverge (power law)

Summary

A phase is a region of analyticity of the free energy per particle $g (T, P)$ ; a phase transition is a locus where $g$ loses analyticity, and the phase diagram partitions the $(T, P)$ plane into lowest- $g$ phases bordered by coexistence curves.
Coexistence requires equal $T$ , $P$ , and chemical potential; the Gibbs phase rule $f = C - Φ + 2$ fixes the dimensionality of each coexistence set, giving areas, curves, and the isolated triple point.
The Clausius-Clapeyron relation $d P / d T = L / (T Δ v)$ sets the slope of a coexistence curve from its latent heat and volume change; the negative slope of the water fusion curve follows from ice being less dense than water.
First-order transitions have a discontinuous first derivative of $g$ : finite latent heat, jumping order parameter, coexistence, and metastability. Continuous transitions have continuous first derivatives, an order parameter vanishing continuously, and diverging response functions.
The Ehrenfest finite-jump picture fails at real critical points, where response functions diverge; the non-analyticity itself exists only in the thermodynamic limit.

Reif, §8.5, and Schroeder, §5.3. The rule counts $C Φ$ chemical potentials less the constraints of equal $μ$ across phases and the Gibbs-Duhem relation within each. ↩