Equations of state and phase behaviour of matter: ideal-gas law, van-der-Waals correction, phase diagrams, the critical point, and the chemical potential that governs phase equilibrium. Upstream of materials science.
phases-and-states
Ideal gas law
Equation of state for an ideal (non-interacting, point-particle) gas: pV = nRT. Accurate at low density and high temperature; deviations…
Van der Waals equation of state
First-order correction to the ideal gas law accounting for finite molecular volume (b) and attractive interactions (a): (p + a n²/V²)(V −…
Phase diagram
Map in (p, T) or (p, T, composition) space of the equilibrium phases of a substance, separated by coexistence curves (melting, boiling,…
Critical point
Endpoint of a liquid-gas coexistence curve where the two phases become indistinguishable. Above T_c the substance is a supercritical fluid.…
Chemical potential (μ)
Partial molar Gibbs free energy μ_i = (∂G/∂n_i)_{T,p,n_{j≠i}}. The driving force for mass transport and phase equilibrium: matter flows…
Triple point
The unique temperature-pressure point on a phase diagram at which the solid, liquid, and gas phases of a substance coexist in thermodynamic…
Vapor pressure
The equilibrium pressure exerted by the vapour of a substance above its condensed (liquid or solid) phase at a given temperature.…
Gibbs phase rule
F = C − P + 2; degrees of freedom = components − phases + 2. Predicts invariant points (triple, eutectic) vs 2D regions in phase diagrams.
Maxwell equal-area construction
Locates liquid/vapor coexistence on vdW loop by equal-area rule: ∫(p-p_eq)dV = 0. Historical foundation for phase equilibrium from…
Supercritical fluids
Above critical point: single homogeneous phase with tunable density/solvent power. sc-CO₂, sc-H₂O (Tc=374°C). Extraction, reactions,…
Glassy state & non-equilibrium
Liquid cooled below Tg vitrifies into amorphous solid; Kauzmann paradox, fragility, dynamic heterogeneity. Physical aging, secondary (β)…
Liquid crystal phases
Nematic (orientational only), smectic (layered), cholesteric (chiral nematic). Displays (TFT-LCD), biological membranes, spider silk.
Eutectic & peritectic points
Eutectic: simultaneous melting of two solids to liquid (lowest T); peritectic: solid₁+L → solid₂ on cooling; phase diagrams.
Spinodal decomposition
Inside spinodal ∂²G/∂c² < 0 gives spontaneous unstable decomposition; interconnected microstructure; Cahn-Hilliard equation.
Ice polymorphs (Ih, II, VI, VII, X)
≥19 ice phases; Ih hexagonal is ambient; high-P proton-ordered phases; ice X at >60 GPa has symmetric H-bonds.
Quasicrystals (Shechtman)
Long-range order without translational periodicity; 5-fold, 10-fold rotations; Al-Mn icosahedral (1984); 2011 Nobel.
Plastic crystals
Orientationally disordered but positionally ordered; CH₄, C₆₀; anomalous heat capacity; rotator phases.
Hume–Rothery solubility rules
≤15% size mismatch, similar crystal structure, electronegativity, valence needed for solid solution; basis of alloy design.
Enantiotropic vs monotropic polymorphism
Transition below melting = enantiotropic (reversible); above = monotropic (irreversible); important in pharmaceutical form selection.
³He and ⁴He superfluids
⁴He BEC below 2.17 K; ³He Cooper pairs in p-wave triplet below 2.6 mK; multiple order parameters, textures.