4.2 Equilibrium, the chemical potential, and stability

The free energy turns “find the equilibrium” into “find the minimum.” This lesson reads three things off the free-energy curve: the equilibrium state from its lowest point, the stability of that state from its curvature, and — when matter can move between phases — the condition that ties coexisting phases together, equality of the chemical potential.

Equilibrium is a minimum

Whatever internal variable the system is free to adjust — an order parameter $x$ , a concentration, a magnetisation, the fraction in one phase — it relaxes until the free energy stops decreasing, $dG/dx = 0$ . The position of that minimum is the equilibrium state. The curvature there decides stability:

\frac{\partial^2 G}{\partial x^2}\bigg|_\text{eq} > 0 \;\Longleftrightarrow\; \text{stable}.

A positive curvature means any small displacement raises the free energy and is pushed back. When the curvature falls to zero — the spinodal — the restoring tendency vanishes, the homogeneous state is no longer even locally stable, and the system separates spontaneously.

ΔG*_A→B0.0625

ΔG*_B→A0.0625

e^−ΔG*/k_BT (A→B)9.39e-1

e^−ΔG*/k_BT (B→A)9.39e-1

Driving force F = 0.000 Temperature k_BT / ΔG*⁰ = 1.00

At F = 0 the wells are symmetric and the barrier is ΔG*⁰ = 1/16 ≈ 0.0625. Pushing F positive tilts the landscape, dropping the B-well, lowering the A→B barrier, and raising the B→A barrier. Beyond F ≈ 0.192 the barrier vanishes entirely — a *spinodal*. The thermal-activation rate goes as Arrhenius's exp(−ΔG*/k_BT): doubling the barrier *squares* the rate.

The landscape carries two wells — two competing states — separated by a barrier. Slide the driving field to tilt it: one well deepens, the other shallows and finally disappears at its spinodal. The lowest well is the equilibrium; a system caught in the higher one is metastable, stable to small perturbations but not the true minimum, the subject of 4.6.

The chemical potential

When the number of particles can change — molecules evaporating from a liquid, atoms crossing from one phase to another — the free energy gains a term for it. The chemical potential is the free-energy cost of adding one particle at fixed temperature and pressure,

\mu \;\equiv\; \left(\frac{\partial G}{\partial N}\right)_{T,p},

so that $dG = -S\,dT + V\,dp + \mu\,dN$ . It is the potential that drives particle flow, exactly as temperature drives heat flow and pressure drives volume change: particles move from high chemical potential to low.

Equilibrium between two phases that can exchange particles therefore requires more than equal temperature and pressure — it requires equal chemical potential. If $\mu_1 > \mu_2$ , moving a particle from phase 1 to phase 2 lowers $G$ , so particles flow until

\mu_1 = \mu_2.

This three-way equality — equal $T$ , equal $p$ , equal $\mu$ — is the condition for phase coexistence, and the starting point for the phase diagram of the next lesson.