Lecture 7 (9/27/2006) Crystal Chemistry Part 6: Phase Diagrams

Gibbs Free Energy G – the energy of a system in excess of its internal energy. (This is the energy necessary for a reaction to proceed) G – the energy of a system in excess of its internal energy. (This is the energy necessary for a reaction to proceed) G = E + PV - TS dG = VdP – SdT at constant T (δG/δP) T = V at constant P (δG/δT) P = -S Stable phases strive to have the lowest G Therefore, the phase with the highest density at a given pressure and the highest entropy at a given temperature will be preferred

Relationship of Gibbs Free Energy to Phase Equilibrium

Clapeyron Equation Defines the state of equilibrium between reactants and product in terms of S and V Defines the state of equilibrium between reactants and product in terms of S and V dG r = V r dP – S r dT dG p = V p dP – S p dT at equilibrium: V r dP – S r dT = V p dP – S p dT or: (V p –V r ) dP = (S p –S r ) dT or: dP/dT = ΔS / ΔV The slope of the equilibrium curve will be positive if S and V both decrease or increase with increased T and P

Reactants -Products V lw < V wv +ΔV V lw < V wv +ΔV S lw < S wv +ΔS S lw < S wv +ΔS Reactants -Products V ice > V lw -ΔV V ice > V lw -ΔV S ice < S lw +ΔS S ice < S lw +ΔS Slope of Phase Reaction Boundaries dP/dT = ΔS / ΔV

Variables Extensive Variables – dependent on the amount of material present Extensive Variables – dependent on the amount of material present mass mass volume volume moles of atoms moles of atoms Intensive Variables – independent on the amount of material present Intensive Variables – independent on the amount of material present pressure pressure temperature temperature density density compositional proportions compositional proportions

Gibbs Phase Rule F = C – Φ + 2 F = C – Φ + 2 F – number of degrees of freedom of intensive variables (p, t, x) that will still preserve chemical equilibrium C – number of components Φ – number of phases

One Component Phase Diagrams IllustratePolymorphismIsochemical P & T are intensive variables Phase Rules: divariant fields F=2 univariant lines F=1 invariant points F=0 Al 2 SiO 5 SiO 2 CaCO 3 C

Two Component Phase Diagrams Solid Solution Crystallization Usually portrayed as isobaric T-X diagrams Usually portrayed as isobaric T-X diagrams For igneous systems, magma/melt is a phase of a simplified composition defined by the mineral phases of interest For igneous systems, magma/melt is a phase of a simplified composition defined by the mineral phases of interest Liquidus – denotes the temperature at which the liquid of a particular compositions will begin to crystallizeLiquidus – denotes the temperature at which the liquid of a particular compositions will begin to crystallize Solidus denotes the temperature at which the liquid of a particular composition will be completely crystallizedSolidus denotes the temperature at which the liquid of a particular composition will be completely crystallized Eutectic Crystallization

DiopsideAnorthite Eutectic Crystallization of Anorthite (plagioclase) and Diopside (pyroxene) Lever Rule Proportions Eutectic Point

Solid Solution Crystallization

Limited Solid Solution and Subsolidus Exsolution: e.g. Alkali Feldspar Increasing Pressure

Exsolution Textures Subsolidus Unmixing Alkali Feldspar Albite Albite exsolution (perthite) Microcline in Microcline host Pyroxene Hypersthene Hypersthene (Opx) exsolution lamellae Augite in Augite (Cpx) host

Multi-component Phase Diagrams Igneous Systems – Liquidus Diagrams LiquidusSurface CotecticLines EutecticPoint

Multi-component Phase Diagrams Metamorphic Systems Chemographic Diagrams e.g. ACF A = Al 2 O 3 +Fe 2 O 3 -Na 2 O-K 2 O C = CaO – 3.3P 2 O 5 F = FeO + MgO + MnO Shows equilibrium assemblages at specified P & T Equilibrium assemblages in metabasalts

Next Lecture 50-minute Test on Crystal Chemistry Lectures 1-7 (see Powerpoints on Website) Klein Chapters 1 (p. 1-16), 3 (p ) and 4 (p ) See CD module 1 for help with ionic coordination Q & A in Lab on tomorrow (Tuesday)