Phase Transitions

A phase transition is an abrupt, discontinuous change in the properties of a system. The stable phase is always the one with lowest free energy $G = E + P V - T S$ .

Why phases exist

At low $T$ : $G$ dominated by $E$ → solid preferred. As $T$ increases, $- T S$ grows → liquid, then gas. Each phase is a tradeoff between energy and entropy.

Van der Waals equation

p = \frac{k_{B} T}{v - b} - \frac{a}{v^{2}}, v = V / N

Two corrections to ideal gas: $b$ is the minimum approach distance between atoms (v cannot go below $b$ ); $a / v^{2}$ captures attractive interactions that reduce pressure.
Isotherms for $T < T_{c}$ develop a wiggle with an unstable region ( $d p / d v > 0$ ). Left branch ( $v \sim b$ ): liquid — dense, nearly incompressible. Right branch: gas — compressible, low density.

Critical point

The wiggle flattens into an inflection point at $T_{c}$ . Condition: $d p / d v = 0$ and $d^{2} p / d v^{2} = 0$ . See Phase Transitions derivations:

k_{B} T_{c} = \frac{8 a}{27 b}, v_{c} = 3 b

Above $T_{c}$ : no phase transition, single fluid. At $T_{c}$ : second order transition ( $S_{liq} \to S_{gas}$ continuously). Below $T_{c}$ : first order transition with latent heat.

Phase equilibrium and Maxwell construction

Two phases coexist when $p$ , $T$ , and $μ$ are equal. Since $G = μ N$ , this means $g_{liq} = g_{gas}$ . Using $d G = V d P$ at constant $T$ , integrating along the isotherm requires:

\int_{loop} v d p = 0

Graphically: the two areas enclosed between the flat line $p = p^{*}$ and the Van der Waals curve must be equal (Maxwell construction). This gives the coexistence pressure. Inside the coexistence curve, isotherms become flat — liquid and gas coexist in whatever proportions keep average density fixed.

Metastable states

Between the spinodal curve (through stationary points of isotherms) and the coexistence curve: states with $d p / d v < 0$ , locally stable but higher $G$ than equilibrium. Supercooled vapour (gas compressed past coexistence) and superheated liquid (liquid expanded past coexistence) — any disturbance triggers condensation/nucleation.

Clausius-Clapeyron equation

On the $p$ - $T$ diagram the coexistence region collapses to a line. Moving along it with $g_{liq} = g_{gas}$ , see Phase Transitions derivations:

\frac{d p}{d T} = \frac{s_{gas} - s_{liq}}{v_{gas} - v_{liq}} = \frac{L}{T (v_{gas} - v_{liq})}

where specific latent heat $L = T (s_{gas} - s_{liq})$ . Slope proportional to latent heat; large when volumes are similar (solid-liquid), small when volumes differ greatly (liquid-gas).

Liquid-gas is a first order transition: $V = \partial G / \partial p$ is discontinuous. At $T_{c}$ it becomes second order.

Latent heat

At the transition $T$ , phases have equal $G$ . Extra heat at constant $T$ converts one phase to the other — temperature does not rise. From $d E = T d S$ at constant $T$ : $E_{2} - E_{1} = T (S_{2} - S_{1})$ .

Symmetry and critical points

Solid-liquid coexistence curve has no critical point: solid has crystal symmetries (translational, rotational) that liquid lacks. Phases with different symmetry groups must always be separated by a transition (Landau symmetry principle). Liquid and gas share the same symmetry group, so one can continuously deform into the other — hence the liquid-gas curve ends at a critical point.

Triple point

Unique $(P, T)$ where solid, liquid, gas all have equal $G$ and coexist. Below triple-point pressure: no liquid phase; solid goes directly to gas (sublimation).