Contributions to Mineralogy and Petrology

, Volume 145, Issue 4, pp 492–501

Activity–composition relations for phases in petrological calculations: an asymmetric multicomponent formulation

Original Paper

Abstract

For petrological calculations, including geothermobarometry and the calculation of phase diagrams (for example, PT petrogenetic grids and pseudosections), it is necessary to be able to express the activity–composition (ax) relations of minerals, melt and fluid in multicomponent systems. Although the symmetric formalism—a macroscopic regular model approach to ax relations—is an easy-to-formulate, general way of doing this, the energetic relationships are a symmetric function of composition. We allow asymmetric energetics to be accommodated via a simple extension to the symmetric formalism which turns it into a macroscopic van Laar formulation. We term this the asymmetric formalism (ASF). In the symmetric formalism, the ax relations are specified by an interaction energy for each of the constituent binaries amongst the independent set of end members used to represent the phase. In the asymmetric formalism, there is additionally a "size parameter" for each of the end members in the independent set, with size parameter differences between end members accounting for asymmetry. In the case of fluid mixtures, for example, H2O–CO2, the volumes of the end members as a function of pressure and temperature serve as the size parameters, providing an excellent fit to the ax relations. In the case of minerals and silicate liquid, the size parameters are empirical parameters to be determined along with the interaction energies as part of the calibration of the ax relations. In this way, we determine the ax relations for feldspars in the systems KAlSi3O8–NaAlSi3O8 and KAlSi3O8–NaAlSi3O8–CaAl2Si2O8, for carbonates in the system CaCO3–MgCO3, for melt in the melting relationships involving forsterite, protoenstatite and cristobalite in the system Mg2SiO4–SiO2, as well as for fluids in the system H2O–CO2. In each case the ax relations allow the corresponding, experimentally determined phase diagrams to be reproduced faithfully. The asymmetric formalism provides a powerful and flexible way of handling ax relations of complex phases in multicomponent systems for petrological calculations.

References

  1. Anovitz LM, Essene EJ (1987) Phase equilibria in the system CaCO3–MgCO3–FeCO3. J Petrol 28:389–414Google Scholar
  2. Anderson GM, Crerar DA (1993) Thermodynamics in Geochemistry. University Press, OxfordGoogle Scholar
  3. Aranovich LY, Newton RC (1999) Experimental determination of CO2–H2O activity–composition relations at 600–1,000 °C and 6–14 kbar by reversed decarbonation and dehydration reactions. Am Mineral 84:1319–1332Google Scholar
  4. Blencoe JG, Seitz JC, Anovitz LM (1999) The CO2–H2O system II. Calculated thermodynamic mixing properties for 400 °C, 0–400 MPa. Geochim Cosmochim Acta 63:2393–2408Google Scholar
  5. Bowen NL, Anderson O (1914) The binary system MgO–SiO2. Am J Sci 37:487–500Google Scholar
  6. Bowen NL, Tuttle OF (1950) The system NaAlSi3O8–KAlSi3O8–H2O. J Geol 58:489–511Google Scholar
  7. Carpenter MA, McConnell JDC (1984) Experimental delineation of the C1̄–I1̄ transformation in intermediate plagioclase feldspars. Am Mineral 69:112–121Google Scholar
  8. Elkins LT, Grove TL (1990) Ternary feldspar experiments and thermodynamic models. Am Mineral 75:544–559Google Scholar
  9. Flowers GC (1979) Correction of Holloway's (1977) adaptation of the Modified Redlich-Kwong equation of state for the calculation of the fugacities of molecular species in supercritical fluids of geological interest. Contrib Mineral Petrol 69:315–318Google Scholar
  10. Fuhrman ML, Lindsley DH (1988) Ternary feldspar modeling and thermometry. Am Mineral 73:201–215Google Scholar
  11. Ghiorso MS (1984) Activity/composition relations in the ternary feldspars. Contrib Mineral Petrol 87:282–296Google Scholar
  12. Goldsmith JR, Newton RC (1969) P–T–X relations in the system CaCO3–MgCO3 at high temperatures and pressures. Am J Sci 267 A:160–190Google Scholar
  13. Goldsmith JR, Newton RC (1974) An experimental determination of the alkali feldspar solvus. In: MacKenzie WS, Zussman J (eds) The Feldspars. University Press, ManchesterGoogle Scholar
  14. Green NL, Usdansky ST (1986) Ternary feldspar relations and feldspar thermobarometry. Am Mineral 71:1100–1108Google Scholar
  15. Greig JW (1927) Liquid immiscibility in the system FeO–Fe2O3–Al2O3–SiO2. Am J Sci 14:473–484Google Scholar
  16. Holland TJB, Powell R (1992) Plagioclase feldspars: activity–composition relations based upon Darken's quadratic formalism and Landau theory. Am Mineral 77:53–61Google Scholar
  17. Holland TJB, Powell R (1996a) Thermodynamics of order–disorder in minerals 1: symmetric formalism applied to minerals of fixed composition. Am Mineral 81:1413–1424Google Scholar
  18. Holland TJB, Powell R (1996b) Thermodynamics of order–disorder in minerals 2: symmetric formalism applied to solid solutions. Am Mineral 81:1425–1437Google Scholar
  19. Holland TJB, Powell R (1998) An internally-consistent thermodynamic data set for phases of petrological interest. J Metamorph Geol 16:309–343Google Scholar
  20. Holloway JR (1977) Fugacity and activity of molecular species in supercritical fluids. In: Fraser DG (ed) Thermodynamics in geology. Reidel, Dordrecht, pp 161–181Google Scholar
  21. Kerrick DM, Jacobs GK (1981) A modified Redlich-Kwong equation for H2O, CO2 and H2O–CO2 mixtures at elevated pressures and temperatures. Am J Sci 281:735–767Google Scholar
  22. Luth WC, Tuttle OF (1966) The alkali feldspar solvus in the system Na2O–K2O–Al2O3–SiO2–H2O. Am Mineral 51:1359–1373Google Scholar
  23. Ol'shanskii YI (1951) Composition of immiscible liquids in volcanic rocks. Contrib Mineral Petrol 80:201–218Google Scholar
  24. Orville PM (1963) Alkali ion exchange between vapor and feldspar phases. Am J Sci 261:201–237Google Scholar
  25. Powell R (1974) A comparison of some mixing models for crystalline silicate solid solutions. Contrib Mineral Petrol 46:265–274Google Scholar
  26. Powell R (1978) Equilibrium thermodynamics in petrology. Harper and Row, New YorkGoogle Scholar
  27. Powell R (1987) Darken's quadratic formalism and the thermodynamics of minerals. Am Mineral 72:1–11Google Scholar
  28. Powell R, Holland TJB (1988) An internally consistent dataset with uncertainties and correlations. 3. Applications to geobarometry, worked examples and a computer program. J Metamorph Geol 6:173–204Google Scholar
  29. Powell R, Holland TJB (1993) On the formulation of simple mixing models for complex phases. Am Mineral 78:1174–1180Google Scholar
  30. Powell R, Holland TJB, Worley B (1998) Calculating phase diagrams with THERMOCALC: methods and examples. J Metamorph Geol 16:577–588Google Scholar
  31. Prausnitz JM, Lichtenthaler RN, de Azevedo EG (1986) Molecular thermodynamics of fluid-phase equilibria, 2nd edn. Prentice-Hall, Englewood CliffsGoogle Scholar
  32. Saxena SK, Fei Y (1988) Fluid mixtures in the C–H–O system at high pressure and temperature. Geochim Cosmochim Acta 52:505–512Google Scholar
  33. Shi P, Saxena SK (1992) Thermodynamic modeling of the C–H–O–S fluid system. Am Mineral 77:1038–1049Google Scholar
  34. Takenouchi S, Kennedy GC (1964) The binary system H2O–CO2 at high temperatures and pressures. Am J Sci 262:1055–1074Google Scholar
  35. Thompson JB Jr, Waldbaum DR (1968a) Mixing properties of sanidine crystalline solutions. I. Calculations based on ion-exchange data. Am Mineral 53:1965–1999Google Scholar
  36. Thompson JB Jr, Waldbaum DR (1968b) Analysis of the two-phase region halite-sylvite in the system NaCl–KCl. Geochim Cosmochim Acta 33:671–690Google Scholar
  37. Thompson JB, Waldbaum DR (1969) Mixing properties of sanidine crystalline solutions. III: calculations based on two phase data. Am Mineral 54:811–838Google Scholar
  38. Todheide K, Franck EV (1963) Das Zweiphasengebiet und die kritische Kurve im System Kohlendioxid–Wasser bis zu Drucken von 3500 bar. Zeitsch Phys Chem 37:387–401Google Scholar
  39. van Laar JJ (1906) Sechs Vorträge über das Thermodynamischer Potential. Vieweg, BrunswickGoogle Scholar
  40. White RW, Powell R, Holland TJB (2001) Calculation of partial melting equilibria in the system Na2O–CaO–K2O–FeO–MgO–Al2O3–SiO2–H2O (NCKFMASH). J Metamorph Geol 19:139–153CrossRefGoogle Scholar
  41. Wood BJ, Banno S (1973) Garnet-orthopyroxene and orthopyroxene-clinopyroxene relationships in simple and complex systems. Contrib Mineral Petrol 42:109–124Google Scholar

Copyright information

© Springer-Verlag 2003

Authors and Affiliations

  1. 1.Department of Earth SciencesUniversity of CambridgeCambridge UK
  2. 2.School of Earth SciencesUniversity of MelbourneVictoria Australia

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