Thermodynamics Begin with a brief review of Chapter 5 Natural systems tend toward states of minimum energy

Energy States Unstable: falling or rolling l Stable: at rest in lowest energy state l Metastable: in low-energy perch Figure 5.1. Stability states. Winter (2010) An Introduction to Igneous and Metamorphic Petrology. Prentice Hall.

Gibbs Free Energy Gibbs free energy is a measure of chemical energy Gibbs free energy for a phase: G = H - TS Where: G = Gibbs Free Energy H = Enthalpy (heat content) T = Temperature in Kelvins S = Entropy (can think of as randomness)

Thermodynamics  G for a reaction of the type: 2 A + 3 B = C + 4 D  G =  (n G) products -  (n G) reactants = G C + 4G D - 2G A - 3G B The side of the reaction with lower G will be more stable

Thermodynamics For other temperatures and pressures we can use the equation: dG = VdP - SdT (ignoring  X for now) where V = volume and S = entropy (both molar) We can use this equation to calculate G for any phase at any T and P by integrating zz GGVdPSdT TPTP T T P P  If V and S are constants, our equation reduces to: G T2 P2 - G T1 P1 = V(P 2 - P 1 ) - S (T 2 - T 1 )

Now consider a reaction, we can then use the equation: d  G =  VdP -  SdT (again ignoring  X)  G for any reaction = 0 at equilibrium

Worked Problem #2 used: d  G =  VdP -  SdT and G, S, V values for albite, jadeite and quartz to calculate the conditions for which  G of the reaction: Ab + Jd = Q is equal to 0 from G values for each phase at 298K and 0.1 MPa calculate  G 298, 0.1 for the reaction, do the same for  V and  S  G at equilibrium = 0, so we can calculate an isobaric change in T that would be required to bring  G 298, 0.1 to  G 298, 0.1 = -  S (T eq - 298)(at constant P) l Similarly we could calculate an isothermal change 0 -  G 298, 0.1 = -  V (P eq - 0.1)(at constant T) Method:

NaAlSi 3 O 8 = NaAlSi 2 O 6 + SiO 2 P - T phase diagram of the equilibrium curve How do you know which side has which phases? Figure Temperature-pressure phase diagram for the reaction: Albite = Jadeite + Quartz calculated using the program TWQ of Berman (1988, 1990, 1991). Winter (2010) An Introduction to Igneous and Metamorphic Petrology. Prentice Hall.

pick any two points on the equilibrium curve d  G = 0 =  VdP -  SdT Thus dP dT S V    Figure Temperature-pressure phase diagram for the reaction: Albite = Jadeite + Quartz calculated using the program TWQ of Berman (1988, 1990, 1991). Winter (2010) An Introduction to Igneous and Metamorphic Petrology. Prentice Hall.

Return to dG = VdP - SdT, for an isothermal process: GGVdP PP P P  z Gas Phases For solids it was fine to ignore V as f(P) For gases this assumption is shitty You can imagine how a gas compresses as P increases How can we define the relationship between V and P for a gas?

Gas Pressure-Volume Relationships Ideal Gas – As P increases V decreases – PV=nRT Ideal Gas Law P = pressure V = volume T = temperature n = # of moles of gas R = gas constant = J mol -1 K -1 P x V is a constant at constant T Figure 5.5. Piston-and-cylinder apparatus to compress a gas. Winter (2010) An Introduction to Igneous and Metamorphic Petrology. Prentice Hall.

Gas Pressure-Volume Relationships Since we can substitute RT/P for V (for a single mole of gas), thus: and, since R and T are certainly independent of P: GGVdP PP P P  z GG RT P dP PP P P  z z GGRT P dP PP P P  1

Gas Pressure-Volume Relationships And since G P2 - G P1 = RT ln P 2 - ln P 1 = RT ln (P 2 /P 1 ) Thus the free energy of a gas phase at a specific P and T, when referenced to a standard atate of 0.1 MPa becomes: G P, T - G T = RT ln (P/P o ) G of a gas at some P and T = G in the reference state (same T and 0.1 MPa) + a pressure term 1 x dxx  z ln o

Gas Pressure-Volume Relationships The form of this equation is very useful G P, T - G T = RT ln (P/P o ) For a non-ideal gas (more geologically appropriate) the same form is used, but we substitute fugacity ( f ) for P where f =  P  is the fugacity coefficient Tables of fugacity coefficients for common gases are available At low pressures most gases are ideal, but at high P they are not o

Dehydration Reactions Mu + Q = Kspar + Sillimanite + H 2 O We can treat the solids and gases separately G P, T - G T =  V solids (P - 0.1) + RT ln (P/0.1) (isothermal) The treatment is then quite similar to solid-solid reactions, but you have to solve for the equilibrium P by iteration

Dehydration Reactions (qualitative analysis) dP dT S V    Figure Pressure-temperature phase diagram for the reaction muscovite + quartz = Al 2 SiO 5 + K- feldspar + H 2 O, calculated using SUPCRT (Helgeson et al., 1978). Winter (2010) An Introduction to Igneous and Metamorphic Petrology. Prentice Hall.

Solutions: T-X relationships Ab = Jd + Q was calculated for pure phases When solid solution results in impure phases the activity of each phase is reduced Use the same form as for gases (RT ln P or ln f) Instead of fugacity, we use activity Ideal solution: a i = X i n = # of sites in the phase on which solution takes place Non-ideal: a i =  i X i where  i is the activity coefficient n n

Solutions: T-X relationships Example: orthopyroxenes (Fe, Mg)SiO 3 – Real vs. Ideal Solution Models Figure Activity-composition relationships for the enstatite-ferrosilite mixture in orthopyroxene at 600 o C and 800 o C. Circles are data from Saxena and Ghose (1971); curves are model for sites as simple mixtures (from Saxena, 1973) Thermodynamics of Rock-Forming Crystalline Solutions. Winter (2010) An Introduction to Igneous and Metamorphic Petrology. Prentice Hall.

Solutions: T-X relationships Back to our reaction: Simplify for now by ignoring dP and dT For a reaction such as: aA + bB = cC + dD At a constant P and T: where:GGRTK PTPT o,,ln K ccDdA a B b  aa aa

Compositional variations Effect of adding Ca to albite = jadeite + quartz plagioclase = Al-rich Cpx + Q  G T, P =  G o T, P + RTlnK Let’s say  G o T, P was the value that we calculated for equilibrium in the pure Na-system (= 0 at some P and T)  G o T, P =  G 298,  V (P - 0.1) -  S (T-298) = 0 By adding Ca we will shift the equilibrium by RTlnK We could assume ideal solution and K Jd Pyx SiO Q Ab Plag  XX X 2 All coefficients = 1

Compositional variations So now we have:  G T, P =  G o T, P + RTln since Q is pure  G o T, P  as calculated for the pure system at P and T  G T, P is the shifted  G due to the Ca added (no longer 0) Thus we could calculate a  V(P - P eq ) that would bring  G T, P back to 0, solving for the new P eq X X Jd Pyx Ab Plag

Compositional variations Effect of adding Ca to albite = jadeite + quartz  G P, T =  G o P, T + RTlnK numbers are values for K P-T phase diagram for the reaction Jadeite + Quartz = Albite for various values of K. The equilibrium curve for K = 1.0 is the reaction for pure end-member minerals (Figure 27.1). Data from SUPCRT (Helgeson et al., 1978). Winter (2010) An Introduction to Igneous and Metamorphic Petrology. Prentice Hall. Figure P-T phase diagram for the reaction Jadeite + Quartz = Albite for various values of K. The equilibrium curve for K = 1.0 is the reaction for pure end-member minerals (Figure 27.1). Data from SUPCRT (Helgeson et al., 1978). Winter (2010) An Introduction to Igneous and Metamorphic Petrology. Prentice Hall.

Geothermobarometry Use measured distribution of elements in coexisting phases from experiments at known P and T to estimate P and T of equilibrium in natural samples

Geothermobarometry The Garnet - Biotite geothermometer

Geothermobarometry The Garnet - Biotite geothermometer Figure Graph of l nK vs. 1/T (in Kelvins) for the Ferry and Spear (1978) garnet-biotite exchange equilibrium at 0.2 GPa from Table Winter (2010) An Introduction to Igneous and Metamorphic Petrology. Prentice Hall. lnK D = · T(K)  G P,T = 0 =  H 0.1, T  S 0.1, P  V + 3 RTlnK D

Geothermobarometry The Garnet - Biotite geothermometer Figure AFM projections showing the relative distribution of Fe and Mg in garnet vs. biotite at approximately 500 o C (a) and 800 o C (b). From Spear (1993) Metamorphic Phase Equilibria and Pressure-Temperature-Time Paths. Mineral. Soc. Amer. Monograph 1.

Geothermobarometry The Garnet - Biotite geothermometer Figure Pressure-temperature diagram similar to Figure 27.4 showing lines of constant K D plotted using equation (27.35) for the garnet- biotite exchange reaction. The Al 2 SiO 5 phase diagram is added. From Spear (1993) Metamorphic Phase Equilibria and Pressure-Temperature- Time Paths. Mineral. Soc. Amer. Monograph 1.

Geothermobarometry The GASP geobarometer Figure P-T phase diagram showing the experimental results of Koziol and Newton (1988), and the equilibrium curve for reaction (27.37). Open triangles indicate runs in which An grew, closed triangles indicate runs in which Grs + Ky + Qtz grew, and half-filled triangles indicate no significant reaction. The univariant equilibrium curve is a best-fit regression of the data brackets. The line at 650 o C is Koziol and Newton’s estimate of the reaction location based on reactions involving zoisite. The shaded area is the uncertainty envelope. After Koziol and Newton (1988) Amer. Mineral., 73,

Geothermobarometry The GASP geobarometer Figure P-T diagram contoured for equilibrium curves of various values of K for the GASP geobarometer reaction: 3 An = Grs + 2 Ky + Qtz. From Spear (1993) Metamorphic Phase Equilibria and Pressure-Temperature-Time Paths. Mineral. Soc. Amer. Monograph

Geothermobarometry

Figure P-T diagram showing the results of garnet-biotite geothermometry (steep lines) and GASP barometry (shallow lines) for sample 90A of Mt. Moosilauke (Table 27.4). Each curve represents a different calibration, calculated using the program THERMOBAROMETRY, by Spear and Kohn (1999). The shaded area represents the bracketed estimate of the P-T conditions for the sample. The Al 2 SiO 5 invariant point also lies within the shaded area. Geothermobarometry

Figure P-T phase diagram calculated by TQW 2.02 (Berman, 1988, 1990, 1991) showing the internally consistent reactions between garnet, muscovite, biotite, Al 2 SiO 5 and plagioclase, when applied to the mineral compositions for sample 90A, Mt. Moosilauke, NH. The garnet-biotite curve of Hodges and Spear (1982) Amer. Mineral., 67, has been added. Geothermobarometry TWQ and THERMOCALC accept mineral composition data and calculate equilibrium curves based on an internally consistent set of calibrations and activity-composition mineral solution models. Rob Berman’s TWQ 2.32 program calculated relevant equilibria relating the phases in sample 90A from Mt. Moosilauke. TWQ also searches for and computes all possible reactions involving the input phases, a process called multi-equilibrium calculations by Berman (1991). Output from these programs yields a single equilibrium curve for each reaction and should produce a tighter bracket of P-T-X conditions.

Figure Reactions for the garnet-biotite geothermometer and GASP geobarometer calculated using THERMOCALC with the mineral compositions from sample PR13 of Powell (1985). A P-T uncertainty ellipse, and the “optimal” AvePT ( ) calculated from correlated uncertainties using the approach of Powell and Holland (1994). b. Addition of a third independent reaction generates three intersections (A, B, and C). The calculated AvePT lies within the consistent band of overlap of individual reaction uncertainties (yet outside the ABC triangle). Geothermobarometry THERMOCALC (Holland and Powell) also based on an internally-consistent dataset and produces similar results, which Powell and Holland (1994) call optimal thermobarometry using the AvePT module. THERMOCALC also considers activities of each end-member of the phases to be variable within the uncertainty of each activity model, defining bands for each reaction within that uncertainty (shaded blue). Calculates an optimal P-T point within the correlated uncertainty of all relevant reactions via least squares and estimates the overall activity model uncertainty. The P and T uncertainties for the Grt-Bt and GASP equilibria are about  0.1 GPa and 75 o C, respectively. A third independent reaction involving the phases present was found (Figure 27.12b). Notice how the uncertainty increases when the third reaction is included, due to the effect of the larger uncertainty for this reaction on the correlated overall uncertainty. The average P-T value is higher due to the third reaction, and may be considered more reliable when based on all three.

Figure P-T pseudosection calculated by THERMOCALC for a computed average composition in NCKFMASH for a pelitic Plattengneiss from the Austrian Eastern Alps. The large + is the calculated average PT (= 650 o C and 0.65 GPa) using the mineral data of Habler and Thöni (2001). Heavy curve through AvePT is the average P calculated from a series of temperatures (Powell and Holland, 1994). The shaded ellipse is the AvePT error ellipse (R. Powell, personal communication). After Tenczer et al. (2006). Geothermobarometry Thermobarometry may best be practiced using the pseudosection approach of THERMOCALC (or Perple_X), in which a particular whole-rock bulk composition is defined and the mineral reactions delimit a certain P-T range of equilibration for the mineral assemblage present. The peak metamorphic mineral assemblage: garnet + muscovite + biotite + sillimanite + quartz + plagioclase + H 2 O, is shaded (and considerably smaller than the uncertainty ellipse determined by the AvePT approach). The calculated compositions of garnet, biotite, and plagioclase within the shaded area are also contoured (inset). They compare favorably with the reported mineral compositions of Habler and Thöni (2001) and can further constrain the equilibrium P and T.

Figure Chemically zoned plagioclase and poikiloblastic garnet from meta-pelitic sample 3, Wopmay Orogen, Canada. a. Chemical profiles across a garnet (rim  rim). b. An-content of plagioclase inclusions in garnet and corresponding zonation in neighboring plagioclase. After St-Onge (1987) J. Petrol. 28, Geothermobarometry P-T-t Paths

Figure The results of applying the garnet-biotite geothermometer of Hodges and Spear (1982) and the GASP geobarometer of Koziol (1988, in Spear 1993) to the core, interior, and rim composition data of St-Onge (1987). The three intersection points yield P-T estimates which define a P-T-t path for the growing minerals showing near-isothermal decompression. After Spear (1993). Geothermobarometry P-T-t Paths

Figure Clockwise P-T-t paths for samples D136 and D167 from the Canadian Cordillera and K98-6 from the Pakistan Himalaya. Monazite U-Pb ages of black dots are in Ma. Small-dashed lines are Al 2 SiO 5 polymorph reactions and large-dashed curve is the H 2 O- saturated minimum melting conditions. After Foster et al. (2004). Geothermobarometry P-T-t Paths Recent advances in textural geochronology have allowed age estimates for some points along a P-T-t path, finally placing the “t” term in “P-T-t” on a similar quantitative basis as P and T. Foster et al. (2004) modeled temperature and pressure evolution of two amphibolite facies metapelites from the Canadian Cordillera and one from the Pakistan Himalaya. Three to four stages of monazite growth were recognized texturally in the samples, and dated on the basis of U-Pb isotopes in Monazite analyzed by LA-ICPMS. Used the P-T-t paths to constrain the timing of thrusting (pressure increase) along the Monashee décollement in Canada (it ceased about 58 Ma b.p.), followed by exhumation beginning about 54 Ma. Himalayan sample records periods of monazite formation during garnet growth at 82 Ma, followed by later monazite growth during uplift and garnet breakdown at 56 Ma, and a melting event during subsequent decompression. Such data combined with field recognition of structural features can elucidate the metamorphic and tectonic history of an area and also place constraints on kinematic and thermal models of orogeny.

Figure An illustration of precision vs. accuracy. a. The shots are precise because successive shots hit near the same place (reproducibility). Yet they are not accurate, because they do not hit the bulls-eye. b. The shots are not precise, because of the large scatter, but they are accurate, because the average of the shots is near the bulls-eye. c. The shots are both precise and accurate. Winter (2010) An Introduction to Igneous and Metamorphic Petrology. Prentice Hall. Geothermobarometry Precision and Accuracy

Figure P-T diagram illustrating the calculated uncertainties from various sources in the application of the garnet-biotite geothermometer and the GASP geobarometer to a pelitic schist from southern Chile. After Kohn and Spear (1991b) Amer. Mineral., 74, and Spear (1993) From Spear (1993) Metamorphic Phase Equilibria and Pressure-Temperature-Time Paths. Mineral. Soc. Amer. Monograph 1. Geothermobarometry Precision and Accuracy