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.
Gibbs phase rule (1876)
J W Gibbs 1876 F = C - P + 2; modern degrees-of-freedom + invariant-points; basis of phase-diagram analysis.
Clausius-Clapeyron (1834-1850)
Clausius 1850 + Clapeyron 1834 dP/dT = Delta-H/(T Delta-V); modern critical-point + supercritical regions.
Eutectic (Roozeboom 1892)
B Roozeboom 1892 eutectic-point binary phase diagram; modern liquid-liquid + solid-liquid + ternary diagrams.
Supercritical (Cagniard 1822)
Cagniard de la Tour 1822 + Andrews 1869 critical-point + supercritical CO2; modern SCF extraction + cleaning.
Ostwald rule (1897)
W Ostwald 1897 step-rule of stages; modern crystallization + nucleation + polymorphism.
Phase field (Cahn-Hilliard 1958)
Cahn-Hilliard 1958 phase-field PDE; modern Allen-Cahn + multi-component + dendritic-solidification.
Gibbs phase rule (1875)
J W Gibbs 1875 F=C-P+2; foundation thermodynamic-phase-equilibria; modern computational-CALPHAD multicomponent prediction.
vdW equation (1873)
J D van der Waals 1873 (Nobel 1910) (P+a/V²)(V-b)=RT; modern equation-of-state + Maxwell-construction + critical-points classification.
Clausius-Clapeyron (1834)
B Clapeyron 1834 + Clausius 1850 dp/dT=L/TΔV phase-boundary; modern modern saturation-vapor-pressure + atmosphere-water-cycle 7%/°C.
LC (Reinitzer 1888)
F Reinitzer 1888 cholesteryl-benzoate-LC; modern de Gennes 1971 nematic + smectic + cholesteric + LC-display 1971+ Grandjean.
Supercritical (Andrews 1869)
T Andrews 1869 critical-point CO2; modern supercritical-CO2 extraction + green-chemistry + scCO2 cleaning + 2024 SF-power Brayton.
Plasma (Langmuir 1928)
I Langmuir 1928 (Nobel 1932) coined-plasma; modern fusion-deuterium-tritium + ITER + cold-atmospheric-plasma medical sterilization.