---
title: Phases, Coexistence, and the Classification of Transitions
module: Phase Transitions
moduleNumber: 11
lessonNumber: 1
order: 1101
summary: >
  A phase transition is a point where the free energy of a substance loses
  analyticity, so a small change in temperature or pressure produces a
  qualitative change of state. This lesson maps the coexistence curves of a pure
  substance, derives the Clausius-Clapeyron relation between the slope of a
  coexistence line and its latent heat, and separates first-order transitions
  (discontinuous entropy and density) from continuous ones (a vanishing order
  parameter and divergent response). The Ehrenfest scheme, the order parameter,
  and the triple and critical points fix the vocabulary the rest of the module
  builds on.
topics: [Phase Transitions]
sources:
  - book: Schroeder
    ref: "Ch. 5 — Free Energy and Chemical Thermodynamics; §5.3 Phase Transitions of Pure Substances"
  - book: Kardar (Statistical Physics of Fields)
    ref: "Ch. 1 — Collective Behavior, from Particles to Fields; §1.1–1.4"
  - book: Pathria & Beale
    ref: "Ch. 12 — Phase Transitions: Criticality, Universality, and Scaling; §12.1"
  - book: Reif
    ref: "Ch. 8 — Equilibrium between Phases and Chemical Species; §8.5"
draft: false
---

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.

> **Definition (Coexistence).** Two phases labeled $1$ and $2$ are in mutual
> equilibrium when their temperatures, pressures, and chemical potentials are all
> equal. For a one-component system the chemical potential equals the Gibbs free
> energy per particle, so the condition is $g_1(T,P) = g_2(T,P)$.

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**.

$$
% caption: The $T$–$P$ phase diagram of a simple substance; the sublimation, fusion, and vaporization curves meet at the triple point, and the vaporization curve terminates at the critical point beyond which liquid and gas are one phase.
\begin{tikzpicture}[>=Latex,font=\footnotesize,scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0,0)--(6.4,0) node[right,black]{$T$};
\draw[->,black] (0,0)--(0,4.6) node[above,black]{$P$};
% triple point at (2,1.3), critical point at (5,3.3)
\draw[acc,thick] (0.35,0.15) .. controls (1.1,0.55) and (1.6,0.95) .. (2,1.3);
\draw[acc,thick] (2,1.3) .. controls (3.0,1.95) and (4.2,2.75) .. (5,3.3);
\draw[acc,thick] (2,1.3) .. controls (2.25,2.3) and (2.5,3.3) .. (2.7,4.3);
\filldraw[black] (2,1.3) circle (2.2pt);
\filldraw[black] (5,3.3) circle (2.2pt);
\node[black!75,below right] at (2.05,1.25) {triple point};
\node[black!75,right] at (5.05,3.3) {critical point};
\node[acc] at (0.9,3.2) {solid};
\node[acc] at (3.1,3.6) {liquid};
\node[acc] at (4.6,0.9) {gas};
\node[black,rotate=34] at (3.5,2.05) {vaporization};
\node[black,rotate=72] at (2.15,3.0) {fusion};
\node[black,rotate=27] at (1.05,0.45) {sublimation};
\end{tikzpicture}
$$

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 $\Phi$ coexisting phases,

$$
f = C - \Phi + 2,
$$

with $f$ the number of degrees of freedom. A single phase of a pure substance
($C=1,\Phi=1$) 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.[^phaserule]

[^phaserule]: Reif, §8.5, and Schroeder, §5.3. The rule counts $C\Phi$ chemical
potentials less the constraints of equal $\mu$ across phases and the
Gibbs-Duhem relation within each.

## 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,

$$
-s_1\,\d T + v_1\,\d P = -s_2\,\d T + v_2\,\d P
\quad\Longrightarrow\quad
\frac{\d P}{\d T} = \frac{s_2 - s_1}{v_2 - v_1} .
$$

The entropy difference is measured by the **latent heat** $L = T\,\Delta s$, the
heat absorbed per particle in converting phase $1$ to phase $2$ at fixed
temperature. Writing $\Delta v = v_2 - v_1$,

> **Clausius-Clapeyron relation.** The slope of a coexistence curve equals the
> latent heat divided by the temperature times the volume change across the
> transition,
> $$
> \frac{\d P}{\d T} = \frac{L}{T\,\Delta v} .
> $$

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.

$$
% caption: The Clausius-Clapeyron slope reads off the coexistence curve; at each point $\d P/\d T$ equals $L/(T\,\Delta v)$, the latent heat over temperature times the volume jump between phases.
\begin{tikzpicture}[>=Latex,font=\footnotesize,scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0,0)--(6.2,0) node[right,black]{$T$};
\draw[->,black] (0,0)--(0,4.4) node[above,black]{$P$};
\draw[acc,thick] (0.6,0.5) .. controls (2.4,1.5) and (3.8,2.6) .. (5.4,4.0);
\coordinate (pt) at (3.3,2.16);
\filldraw[black] (pt) circle (2.0pt);
% tangent line at pt, slope ~0.86
\draw[black,thick] (2.0,1.03) -- (4.6,3.29);
\draw[black,dashed] (4.0,2.76) -- (4.6,2.76) -- (4.6,3.29);
\node[black,right] at (4.62,3.02) {$dP$};
\node[black,below] at (4.3,2.74) {$dT$};
\node[acc] at (1.2,3.4) {phase 1};
\node[acc] at (4.7,1.1) {phase 2};
\node[black,below right] at (2.35,1.02) {tangent line};
\end{tikzpicture}
$$

> **Worked example.** The fusion curve of water slopes the wrong way. Ice is less
> dense than liquid water, so on melting the volume per molecule decreases,
> $\Delta v = v_{\text{liq}} - v_{\text{sol}} < 0$, while the latent heat of
> fusion is positive, $L > 0$. The Clausius-Clapeyron relation then gives $\d P/\d
> T < 0$: the melting curve tilts up and to the left, and raising the pressure
> lowers the melting temperature. Numerically, $L \approx 6.0\ \mathrm{kJ\,
> mol^{-1}}$, $\Delta v \approx -1.6\times 10^{-6}\ \mathrm{m^3\,mol^{-1}}$, and
> $T = 273\ \mathrm{K}$ give $\d P/\d T \approx -1.4\times 10^{7}\ \mathrm{Pa\,
> K^{-1}}$, so depressing the melting point by one kelvin takes about
> $140\ \mathrm{atm}$. Nearly every other substance has $\Delta v > 0$ on
> melting and a fusion curve that leans the opposite way.

For vaporization the volume change is dominated by the gas, $\Delta v \approx
v_{\text{gas}} \approx k_B T/P$ if the vapor is treated as ideal, and the latent
heat is nearly constant over a modest range. Substituting turns the relation into
$\d P/\d T \approx LP/(k_B T^2)$, 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.

$$
% caption: At a first-order transition the physical free energy is the lower envelope of two analytic branches; its slope (the entropy) jumps at the crossing, and the entropy versus temperature shows a step of height $L/T$.
\begin{tikzpicture}[>=Latex,font=\footnotesize,scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
% left panel: g vs T
\draw[->,black] (0,0)--(4.2,0) node[right,black]{$T$};
\draw[->,black] (0,0.2)--(0,3.6) node[above,black]{$g$};
\draw[acc,thick] (0.4,3.2) .. controls (1.6,2.0) and (2.1,1.55) .. (2.4,1.42);
\draw[acc,thick,dotted] (2.4,1.42) .. controls (2.9,1.2) and (3.4,1.0) .. (3.9,0.82);
\draw[black,thick] (2.4,1.42) .. controls (3.0,0.8) and (3.5,0.4) .. (3.9,0.18);
\draw[black,thick,dotted] (2.4,1.42) .. controls (2.05,1.75) and (1.7,2.1) .. (1.3,2.55);
\filldraw[black] (2.4,1.42) circle (1.6pt);
\node[acc,above right] at (0.5,3.0) {phase 1};
\node[black,below] at (3.6,0.35) {phase 2};
\node[black,below] at (2.4,1.3) {$T_t$};
% right panel: s vs T with a jump
\begin{scope}[xshift=5.4cm]
\draw[->,black] (0,0)--(4.2,0) node[right,black]{$T$};
\draw[->,black] (0,0.2)--(0,3.6) node[above,black]{$s$};
\draw[acc,thick] (0.4,0.6) -- (2.4,1.5);
\draw[black,thick] (2.4,2.7) -- (3.9,3.35);
\draw[black,dashed] (2.4,1.5)--(2.4,2.7);
\node[black,below] at (2.4,-0.02) {$T_t$};
\node[black,right] at (2.45,2.1) {$\dfrac{L}{T_t}$};
\end{scope}
\end{tikzpicture}
$$

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 $\kappa_T = -v^{-1}(\partial
v/\partial P)_T$ 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
  $\rho_{\text{liq}} - \rho_{\text{gas}}$, 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.

$$
% caption: Across a continuous transition the order parameter rises from zero as the temperature drops through $T_c$, vanishing with a characteristic power law; above $T_c$ it is identically zero.
\begin{tikzpicture}[>=Latex,font=\footnotesize,scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0,0)--(6.2,0) node[right,black]{$T$};
\draw[->,black] (0,0)--(0,3.4) node[above right,black]{order parameter};
% Tc at x=4.2; below Tc the curve rises to the left as (Tc-T)^{1/2}
\draw[acc,very thick] (4.2,0) .. controls (3.6,1.35) and (2.6,2.1) .. (0.4,2.75);
\draw[black,very thick] (4.2,0) -- (6.0,0);
\filldraw[black] (4.2,0) circle (1.8pt);
\node[black,below] at (4.2,-0.05) {$T_c$};
\node[acc] at (1.7,2.55) {ordered};
\node[black,above] at (5.1,0.05) {disordered};
\end{tikzpicture}
$$

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 ($C$, $\kappa$, $\chi$) |
| 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 $f = C-\Phi+2$ fixes the dimensionality of each coexistence set, giving
  areas, curves, and the isolated triple point.
- The Clausius-Clapeyron relation $\d P/\d T = L/(T\,\Delta v)$ 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.