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Thermobarometry Lecture 12 We now have enough thermodynamics to put it to some real use: calculating the temperatures and pressures at which mineral assemblages (i.e., rocks) equilibrated within the Earth. Some theoretical considerations • We have seen that which phase assemblage is stable and the composition of those phases depends on ∆Gr, which we use to calculate K o We also know ∆Gr depends on T and P. • Reactions that make good geothermometers are those that depend strongly on T. -∆ H ∆ S • ln K = + o r RT R o What would characterize a good geothermometer? • Similarly, a good geobarometer would be one strongly depending on P o ∆ VT,Pref æ ¶ ln K ö çè ÷ø = ¶P T RT • A good geothermometer will have large ∆H; a good geobarometer will have large ∆V. Univariant Reactions • Univariant (or invariant) reactions provide possible thermobarometers. • There are 3 phases in the Al2Si2O5 system. o When two coexist, we need only specify either T or P, the other is then fixed. o All three can coexist at just one T and P. o First is rare, second is rarer. Garnet Peridotite Geobarometry • • • • • • • Garnet becomes the high pressure aluminous phase in the mantle, replacing spinel. Aluminum also dissolves in the orthopyroxene (also clinopyroxene) We can write the reaction as: Mg2Si2O6+MgAl2SiO6 = Mg3Al2Si3O12 l.h.s. is the opx solid solution - Al end member does not exist as pure phase. Significant volume change associated with this reaction (but also depends on T). Other complexities arise from Ca, Fe, and Cr in phases. Original approach of Wood and Banno generally assumed ideal solution Garnet Peridotite Geobarometry • Subsequent refinements used asymmetric solution model to match experimental data. • Recognize two distinct sites in opx crystal: o Smaller M1: Al substitutes here o Larger M2: Ca substitutes here • P given by -C2 C22 + C3C1 P= C3 • where C3 is constant and other parameters depend on K, T, and composition. Gt 3 (1- XCa ) K = M1 M 2 M1 M1 X Mg,Fe X Mg,Fe X Mg,Fe X Al Solvus Equilibria • Another kind of thermobarometer is based on exsolution of two phases from a homogenous single phase solution. • This occurs when the excess free energy exceeds the ideal solution term and inflections develop, as in the alkali feldspar system. • Because it is strongly temperature dependent and not particularly pressure dependent, this makes a good geothermometer. Temperature in Peridotites Ca2+ • Temperatures calculated from compositions of co-existing orthopyroxene (enstatite) and clinopyroxene (diopside) solid solutions, which depend on T. Exchange Reactions • There are a number of common minerals where one or more ions substitutes for others in a solid solution. o The Fe2+–Mg2+ substitution is common in ferromagnesian minerals. • Let’s consider the exchange of Mg and Fe between olivine and a melt containing Mg and Fe. o o This partitioning of these two ions between melt and olivine depends on temperature. We can use a electron microprobe to measure the composition of olivine and co-existing melt (preserved as glass). Olvine-Melt Geothermometer • Reaction of interest can be written as: MgOol + FeOl = MgOl + FeOol o (note, this does not involve redox, so we write it in terms of oxides since these are conventionally reported in analyses. We could write it in terms of ions, however.) • Assuming both solid and liquid solutions are ideal, the equilibrium constant for this reaction is: Ol X FeO X MgO K= Ol X FeO X MgO • Unfortunately ∆H for the reaction above is small, so it has weak temperature dependence. Roeder & Emslie Geothermometer • Roeder & Emslie (1970) decided to consider two separate reactions: • MgOliq –> MgOOl and FeOliq –> FeOOl • Based on empirical data, they deduced the temperature dependence as: See Example 4.3 for how to do the calculation - biggest effort is simply converting wt. percent to mole fraction. Ol X MgO 3740 log K D = log l = -1.87 X MgO T • and Ol XFeO 3911 log K D = log l = - 2.50 XFeO T Buddington and Lindsley Oxide Geothermometer Recall this diagram from Chapter 3 • Things get interesting in real systems containing Ti, because both magnetite and hematite are solid solutions. • Partition of Fe and Ti between the two depends on T and ƒO2. Buddington and Lindsley Oxide Geothermometer • The reaction of interest is: yFe2TiO4 + (1-y)Fe3O4 + ¼O2 = yFeTiO3 + (3/2 -y)Fe2O3 magnetite s.s. hematite s.s. • The equilibrium constant for this reaction is K= y 3/2-y aFeTiO a Fe 3 2O3 y 1-y aFe a ƒO2 TiO 2 4 Fe3O4 • The reaction can be thought of as a combination of an exchange reaction: Fe3O4 + FeTiO3 = Fe3TiO4 + Fe2O3 magnetite + illmenite = ulvospinel + hematite • plus the oxidation of magnetite to hematite: 4Fe3O4 + O2 = 6Fe2O3 Computing Temperature and Oxygen Fugacity • The calculation is complex because the system cannot be treated as ideal (except titanomagnetite above 800˚C). Equilibrium constant is: ∆ G é X (1- X ) ù é l l ù 2 - • and = ln ê 2 êë (1- XUsp )X Ill Usp RT - Ill 2 Hem 2 Mt Ill ú + ln ê úû ë l l Usp ú û 12 é X 12 ù é l Hem ù ∆G = ln ê Hem + ln - ln ƒO2 ê 4 ú 4 ú RT ë X Mt û ë lMt û • Must calculate λ’s using asymmetric solution model (using interaction parameters), then solve for T and ƒO2. Example 4.4 shows how.