Free energy & phase equilibria

Why free energy is the right potential — total-entropy reasoning, Legendre transforms, nucleation, Clausius–Clapeyron, two-state gating.

The thermodynamics chapter gives the rules of bookkeeping — the first law, the equation of state, the adiabatic relation. This chapter gives the rules of prediction: free energy is the function whose minimum identifies the equilibrium state, whose curvature identifies stability, and whose finite-temperature shape governs the rates at which the system hops between competing wells.

Why a free energy is needed — the total-entropy argument

The second law says the entropy of an isolated system never decreases. But most of the systems we care about — a bubble in water, a gating channel in a hair cell, a slab of air in a sound wave — are not isolated; they exchange heat with their surroundings. The relevant criterion shifts.

Imagine the system in thermal contact with a much-larger reservoir at fixed temperature $T$. The combined “system + reservoir” is isolated, so its total entropy never decreases. The reservoir’s entropy change is fixed by the heat it absorbs: $\Delta S_\text{res} = -Q_\text{sys}/T = -\Delta U_\text{sys}/T$ at fixed volume (no work). The total entropy change is therefore

where $F \equiv U - TS$. The second law $\Delta S_\text{tot} \ge 0$ is therefore equivalent to $\Delta F \le 0$. The Helmholtz free energy $F$ is not a new physical quantity — it is the system-level bookkeeping that absorbs the reservoir contribution.

Slide $\Delta U$ and $\Delta S_\text{sys}$. The total-entropy verdict (“Allowed” / “Forbidden”) flips at exactly the $\Delta F = 0$ line — $\Delta F < 0$ is allowed, $\Delta F > 0$ forbidden. That is why we minimise $F$ at fixed $T$ and $V$.

For a system at fixed $T$ and $p$ — the more common case for liquids and gases under atmospheric conditions — the analogous bookkeeping gives the Gibbs free energy

with the criterion $\Delta G \le 0$. The right potential is dictated by which variables are clamped; each is a Legendre transform of $U$.

What a Legendre transform actually does

$F = U - TS$ and $G = U + pV - TS$ look algebraically arbitrary until you see what they are: each is the Legendre transform of $U$, swapping a “natural” variable for its conjugate. $U(S, V, N)$ has natural variables $S, V, N$; differentiating gives $T = \partial U/\partial S$, $p = -\partial U/\partial V$, etc.

The Legendre transform $F(T, V, N) = U - TS$ trades the natural variable $S$ for its conjugate $T$. Geometrically: $F(T)$ is the y-intercept of the tangent to $U(S)$ with slope $T$.

Slide $T$ — the slope of the tangent line. The tangent point on $U(S)$ moves, and the tangent’s y-intercept is $F(T) = U(S^*) - T\,S^*$. This is the same information as $U(S)$, repackaged. The “obvious” reason we use $F$ instead of $U$ is that at fixed temperature, $T$ is the natural control variable — exactly what the experimenter fixes — and the Legendre transform makes the math reflect that.

In differential form,

Reading off natural derivatives,

Equilibrium = minimum, stability = curvature

The equilibrium criterion $dF \le 0$ (or $dG \le 0$) means the system relaxes toward a minimum of the free energy, parametrised by whatever internal variable is free to adjust — a bubble radius, a magnetisation, a chemical concentration, an order parameter $x$. The position of the minimum identifies the equilibrium state. The curvature of the free energy at that point identifies its stability:

When the curvature vanishes — the spinodal — small perturbations no longer have a restoring tendency, and the homogeneous state ceases to exist; the system spontaneously demixes or phase-separates.

Two competing wells: nucleation

When two phases — say liquid and vapour — coexist, the free energy has two minima, each describing one phase, separated by a barrier representing the high-energy intermediate state (a microscopic bubble, a microscopic crystal). The barrier height $\Delta G^*$ controls the rate of spontaneous phase change: by the kinetic theory of activated processes, the rate is proportional to the Boltzmann factor $e^{-\Delta G^*/k_B T}$.

For a spherical bubble of radius $R$ nucleating inside a liquid under tension, two contributions to $G$ compete:

The cubic term dominates at large $R$, the quadratic at small $R$, producing a barrier at the critical radius $R^* = 2\sigma/\Delta p$ with height

Slide $\sigma$ and the driving tension; the volume curve (cubic, growing downward) and surface curve (quadratic, growing upward) sum to a barrier whose height $\Delta G^*$ is read off directly. The Boltzmann factor $e^{-\Delta G^*/k_BT}$ at the bottom shows how astronomically suppressed homogeneous nucleation is: for pure water at room temperature the barrier is hundreds to thousands of $k_BT$, putting the rate so far below any observable level that real cavitation has to nucleate at impurities (the Cavitation Ch 2.2 heterogeneous story).

A general landscape with a tunable driving force

The bubble-nucleation case is one instance of a universal structure. Any system with two competing states has a free-energy landscape

where $G_\text{wells}$ has two minima at $x = 0$ (phase A) and $x = 1$ (phase B), and $F$ is a driving “field” — a pressure difference, a voltage bias, a mechanical force — that tilts the landscape.

Slide $F$ from zero. The landscape tilts; the A→B barrier shrinks; the B→A barrier grows. Beyond a critical $F$ the A-well disappears altogether — a spinodal. Slide $T$ to see the Boltzmann rate factor $e^{-\Delta G^*/k_B T}$ change: at low $T$ the barrier is essentially uncrossable; at high $T$ the two wells equilibrate freely.

Phase equilibrium and the Clausius–Clapeyron equation

At coexistence between two phases — say liquid (l) and vapour (v) — the Gibbs free energies per unit mass are equal: $g_l(p, T) = g_v(p, T)$. As you move along the coexistence curve in the $p$–$T$ plane, the two free energies must change together:

The exponential dependence on $1/T$ is steep: water’s vapour pressure rises from 0.6 kPa at 0°C to 100 kPa at 100°C — a factor of 165 across a 100 K range. Slide the latent heat to see how the curvature depends on $L$.

The same formula governs the vapour pressure inside the bubble of Cavitation 3.2; the body-temperature value (37°C, 6.3 kPa) sets the operating point for cavitation in biological fluids.

Two-state free-energy gating

A clean instance of free-energy reasoning: a channel that switches between an open (O) and closed (C) state, with free-energy difference $\Delta G = G_O - G_C$ that depends linearly on some external bias $F$:

Treating the two states as Boltzmann-weighted, the open probability is

This is the Fermi function in $F$ — sigmoidal, centred at $F = \Delta G_0/\alpha$, with width set by $k_B T/\alpha$.

Slide the bias $F$: the channel opens as $F$ crosses the midpoint. Lower $T$: the transition sharpens; in the $T \to 0$ limit it becomes a step. Raise $\alpha$ (stronger coupling): the transition steepens at fixed $T$. The dimensionless ratio $(\Delta G_0 - \alpha F)/k_BT$ — not any of those quantities alone — determines the open probability.

This is the operative model of mechanotransduction in the inner ear’s hair cells, of voltage-gated ion channels (where $F$ is the membrane voltage), and of any thermodynamic two-state system with a linear external bias.

For the cross-book applications — Cavitation’s homogeneous and heterogeneous nucleation, the MET-channel Fermi function in the cochlea, vapour pressure inside an oscillating bubble — see the key examples sub-page.