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

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

Energy States l Unstable: falling or rolling l Stable: at rest in lowest energy state l Metastable: in low-energy perch Figure 5-1. Stability states. Winter (2001) 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 = 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 211 1 2 1 2 2  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 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 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 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 0  G at equilibrium = 0, so we can calculate an isobaric change in T that would be required to bring  G 298, 0.1 to 0 0 -  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 27-1. Temperature- pressure phase diagram for the reaction: Albite = Jadeite + Quartz calculated using the program TWQ of Berman (1988, 1990, 1991). Winter (2001) 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 27-1. Temperature- pressure phase diagram for the reaction: Albite = Jadeite + Quartz calculated using the program TWQ of Berman (1988, 1990, 1991). Winter (2001) An Introduction to Igneous and Metamorphic Petrology. Prentice Hall.

Return to dG = VdP - SdT, for an isothermal process: GGVdP PP P P 21 1 2  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?

Ideal Gas u As P increases V decreases u PV=nRT Ideal Gas Law F P = pressure F V = volume F T = temperature F n = # of moles of gas F R = gas constant = 8.3144 J mol -1 K -1 = 8.3144 J mol -1 K -1 P x V is a constant at constant T Gas Pressure-Volume Relationships Figure 5-5. Piston-and-cylinder apparatus to compress a gas. Winter (2001) An Introduction to Igneous and Metamorphic Petrology. Prentice Hall.

Gas Pressure-Volume Relationships Since we can substitute RT/P vor V (for a single mole of gas), thus: we can substitute RT/P vor V (for a single mole of gas), thus: and, since R and T are certainly independent of P: GGVdP PP P P 21 1 2  z GG RT P dP PP P P 21 1 2  z z GGRT P dP PP P P 21 1 2  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 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 equuation is very useful G P, T - G T = RT ln (P/P o ) 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 l Mu + Q = Kspar + Sillimanite + H 2 O l We can treat the solids and gases separately G P, T - G T =  V solids (P - 0.1) + RT ln (P/0.1) (isothermal) l 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 27-2. 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 (2001) 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 u Real vs. Ideal Solution Models Figure 27-3. 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 (2001) 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 + RT l nK 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, 0.1 +  V (P - 0.1) -  S (T-298) = 0  G o T, P =  G 298, 0.1 +  V (P - 0.1) -  S (T-298) = 0 By adding Ca we will shift the equilibrium by RT l nK We could assume ideal solution and K JdPyxSiOQAb Plag  XX X 2 All coefficients = 1

Compositional variations So now we have:  G T, P =  G o T, P + RT l n since Q is pure  G o T, P  as calculated for the pure system at P and T  G T, P is the shifed  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 JdPyxAb Plag

Effect of adding Ca to albite = jadeite + quartz  G P, T =  G o P, T + RT l nK Compositional variations numbers are values for K Figure 27-4. 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 (2001) An Introduction to Igneous and Metamorphic Petrology. Prentice Hall.

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

Figure 27-5. 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 27-2. Winter (2001) An Introduction to Igneous and Metamorphic Petrology. Prentice Hall.Geothermobarometry lnK D = -2108 · T(K) + 0.781  G P,T = 0 =  H 0.1, 298 - T  S 0.1, 298 + P  V + 3 RTlnK D

The Garnet - Biotite geothermometer Figure 27-6. 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 27-7. 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 27-8. 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, 216-233Geothermobarometry

The GASP geobarometer Figure 27-8. 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 1.Geothermobarometry

Geothermobarometry

Figure 27-10. 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 27-11. 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, 1118-1134 has been added.Geothermobarometry

Figure 27-12. 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, 1-22.Geothermobarometry P-T-t Paths

Figure 27-13. 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 27-14. 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 (2001) An Introduction to Igneous and Metamorphic Petrology. Prentice Hall.Geothermobarometry Precision and Accuracy

Figure 27-15. 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, 77-84 and Spear (1993) From Spear (1993) Metamorphic Phase Equilibria and Pressure-Temperature-Time Paths. Mineral. Soc. Amer. Monograph 1.Geothermobarometry Precision and Accuracy

Figure 27-16. Phase diagram for the reaction: calcite + quartz = wollastonite + CO 2 calculated using the program SUPCRT, assuming p CO2 = P Lith. Winter (2001) An Introduction to Igneous and Metamorphic Petrology. Prentice Hall.

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