12.109 Lecture Notes September 29, 2005 Thermodynamics II Phase diagrams and exchange reactions Handouts: using phase diagrams, from 12.104, Thermometry and Barometry Fractional crystallization vs. equilibrium crystallization Perfect equilibrium – constant bulk composition, crystals + melt react, reactions go to equilibrium Perfect fractional – situation where reaction between phases is incomplete, melt entirely removed, etc. Because earth is not in equilibrium, we have interesting geology! Binary system = 2 component system Example: albite (Ab) and anorthite (An) solid solution Phase diagrams show the equilibrium case. Fractional crystallization would result in zoned crystal growth: The exchange of ions happens by solid state diffusion. If the crystal grows faster than ions can diffuse through it, the outer layers form with different compositions, thus we have a chemically zoned crystal. Olivine and plagioclase commonly grow this way. In crossed polarized light, you can see gradual extinction from the center, out to the edges of the crystal. This is also sometimes visible in clinopyroxene. Fractional crystallization preserves the original composition. The center zone has the composition of the crystal from the liquidus. The liquidus composition reveals the temperature of the liquid when it arrived at the final crystallization.Solvus, or miscibility gap – in system with solid solution, region of immiscibility (inability to mix) In Na-K feldspars, perthite results from unmixing of a single crystalline phase  two coexisting phases with different compositions and same crystal structure As T goes down, two phases separate out (spinoidal decomposition) Feldspar system, see Bowen and Tuttle Thermometry and Barometry Thermobarometer Igneous and metamorphic rocks Uses composition of coexisting minerals to tell us something about T + P Liquidus minerals record temperature (if you can preserve the composition of the liquidus mineral. Lots of basalts have olivine as a liquidus phase – so if you know the bulk composition of the basalt and the composition of the olivine, you can estimate the crystallization temperature. You need a rock that contains only a few % crystals, ideally 1-2% (a “sparsely phyric” rock).You can write an expression that qualifies the shape of the 2 phase region: an EXCHANGE REACTION – reaction that describes the distribution of a pair of elements between two phases. In igneous systems mineral-melt interactions can be very useful. 1. can test for equilibrium between crystal and melt that it coexists with 2. can be used along with formation reactions to infer emplacement/eruption conditions formation reactions – assembling solid form components in melt examples: olivine formation reactions 1. forsterite-melt 2MgOmelt + SiO2melt  Mg2SiO4oliv 2. fayalite-melt 2FeOmelt + SiO2melt Fe2SiO4oliv exchange reaction: 3. Mg2SiO4oliv + 2FeOmelt Fe2SiO4oliv + 2MgOmeltNote: reaction 2 – reaction 1 = reaction 3 We can write an equilibrium constant: eqkRTG ln−=Δ G = Gibbs free energy R = thermodynamic constant T = temperature k = equilibrium constant )()(2422 meltSiOmeltMgOolivSiOMgeqaaak = a = X * γ For an ideal solution, a = X = mole fraction = (wt%/gram formula weight) renormalized. We approximate that γ = 1, a ≈ X for non-ideal cases. We could write an analogous reaction for fayalite (Fe2SiO4). Note: this is ln K vs. 1/T relationship:The ΔHrxn for these formation reactions is heat of fusion. This has been measured for both fayalite and forsterite. forsteritefusionHΔ ∼25 kcal/mole fayalitefusionHΔ ∼29 kcal/mole Both formation reactions are T dependent and can be used to predict T. The exchange reaction 242422()ln ln ln()oliv meltFe SiO MgOfayalite melt forsterite meltD formation formationoliv meltMg SiO FeOaaKKKaa−−==−∼constant And 0.29-0.33 for basaltic magmas Fe MgDolivK−∼ We can make simplifying assumptions: melt meltMgO MgOaX= and 24olivMg SiO forsteriteaX= melt meltFeO FeOaX= 24olivFe SiO fayaliteaX= %40.311%%[40.311 71.846meltmeltmelt meltMgOwt MgOXwt MgO wt FeO=+ Other useful exchange and formation reactions are: Plagioclase-melt (Ca-Na) – P, H2Oin melt sensitive Pyroxene-melt (Fe-Mg) – T insensitive