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On the nature of the excess heat capacity of mixing

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Abstract

The excess vibrational entropy (ΔS exvib ) of several silicate solid solutions are found to be linearly correlated with the differences in end-member volumes (ΔV i ) and end-member bulk moduli (Δκ i ). If a substitution produces both, larger and elastically stiffer polyhedra, then the substituted ion will find itself in a strong enlarged structure. The frequency of its vibration is decreased because of the increase in bond lengths. Lowering of frequencies produces larger heat capacities, which give rise to positive excess vibrational entropies. If a substitution produces larger but elastically softer polyhedra, then increase and decrease of mean bond lengths may be similar in magnitude and their effect on the vibrational entropy tends to be compensated. The empirical relationship between ΔS exvib , ΔV i and Δκ i , as described by ΔS exvib  = (ΔV i  + mΔκ i )f, was calibrated on six silicate solid solutions (analbite–sanidine, pyrope–grossular, forsterite–fayalite, analbite–anorthite, anorthite–sanidine, CaTs–diopside) yielding m = 0.0246 and f = 2.926. It allows the prediction of ΔS exvib behaviour of a solid solution based on its volume and bulk moduli end-member data.

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References

  • Angel RJ (1994) Feldspars at high pressures. In: Parsons I (ed) Feldspars and their reactions. Kluwer Academic Publishers, Dordrecht, pp 271–312

    Google Scholar 

  • Antonioli G, Bini S, Lottici PP, Razzetti C (1986) The local-structure in the mixed chalcopyrite CuGaXIn1-xSe2. J Phys 47:431–434

    Google Scholar 

  • Badger RM (1934) A relation between internuclear distances and bond force constants. J Chem Phys 2:128–131

    Article  Google Scholar 

  • Benisek A, Kroll H, Cemič L, Kohl V, Breit U, Heying B (2003) Enthalpies in (Na, Ca) and (K, Ca)-feldspar binaries: a high temperature solution calorimetric study. Contrib Mineral Petrol 145:119–129

    Article  Google Scholar 

  • Benisek A, Etzel K, Cemic L (2007) Thermodynamic mixing behavior of synthetic Ca-Tschermak-diopside pyroxene solid solutions: II. Heat of mixing and activity–composition relationships. Phys Chem Miner 34:747–755

    Article  Google Scholar 

  • Benisek A, Dachs E, Kroll H (2009) Excess heat capacity and entropy of mixing in the plagioclase solid solution. Am Mineral 94:1153–1161

    Article  Google Scholar 

  • Benisek A, Dachs E, Kroll H (2010a) Excess heat capacity and entropy of mixing in the high-structural state (K, Ca)-feldspar binary. Phys Chem Miner 37:209–218

    Article  Google Scholar 

  • Benisek A, Dachs E, Kroll H (2010b) Excess heat capacity and entropy of mixing in ternary series of high-structural state feldspars. Eur J Mineral 22:403–410

    Article  Google Scholar 

  • Burton BP, van de Walle A (2006) First-principles phase diagram calculations for the system NaCl–KCl: the role of excess vibrational entropy. Chem Geol 225:222–229

    Article  Google Scholar 

  • Chopelas A (2006) Modeling the thermodynamic parameters of six endmember garnets at ambient and high pressures from vibrational data. Phys Chem Miner 33:363–376

    Article  Google Scholar 

  • Chung DH (1971) Elasticity and equations of state of olivines in the Mg2SiO4–Fe2SiO4 system. Geophys J R Astron Soc 25:511–538

    Google Scholar 

  • Collins MD, Brown JM (1998) Elasticity of an upper mantle clinopyroxene. Phys Chem Miner 26:7–13

    Article  Google Scholar 

  • Dachs E, Geiger CA (2006) Heat capacities and entropies of pyrope-grossular (Mg3Al2Si3O12–Ca3Al2Si3O12) garnet solid solutions: a low-temperature calorimetric and a thermodynamic investigation. Am Mineral 91:894–906

    Article  Google Scholar 

  • Dachs E, Geiger CA, von Seckendorff V, Grodzicki M (2007) A low-temperature calorimetric study of synthetic (forsterite plus fayalite) (Mg2SiO4 + Fe2SiO4) solid solutions: an analysis of vibrational, magnetic, and electronic contributions to the molar heat capacity and entropy of mixing. J Chem Thermodyn 39:906–933

    Article  Google Scholar 

  • Decker DL (1971) High-pressure equation of state for NaCl, KCl and CsCl. J Appl Phys 42:3239–3244

    Article  Google Scholar 

  • Etzel K, Benisek A (2008) Thermodynamic mixing behavior of synthetic Ca-Tschermak-diopside pyroxene solid solutions: III. An analysis of IR line broadening and heat of mixing behavior. Phys Chem Miner 35:399–407

    Article  Google Scholar 

  • Etzel K, Benisek A, Dachs E, Cemic L (2007) Thermodynamic mixing behavior of synthetic Ca-Tschermak-diopside pyroxene solid solutions: I. Volume and heat capacity of mixing. Phys Chem Miner 34:733–746

    Article  Google Scholar 

  • Fiquet G, Guyot F, Itie JP (1994) High-pressure X-ray diffraction study of carbonates: MgCO3, CaMg(CO3)2, and CaCO3. Am Mineral 79:15–23

    Google Scholar 

  • Galoisy L (1996) Local versus average structure around cations in minerals from spectroscopic and diffraction measurements. Phys Chem Miner 23:217–225

    Article  Google Scholar 

  • Geiger CA (1998) A powder infrared spectroscopic investigation of garnet binaries in the system Mg3Al2Si3O12–Fe3Al2Si3O12–Mn3Al2Si3O12–Ca3Al2Si3O12. Eur J Mineral 10:407–422

    Google Scholar 

  • Geiger CA (1999) Thermodynamics of (Fe2+, Mn2+, Mg, Ca)3Al2Si3O12 garnet: an analysis and review. Mineral Petrol 66:271–299

    Article  Google Scholar 

  • Geiger CA (2001) Thermodynamic mixing properties of binary oxide and silicate solid solutions determined by direct measurements: the role of strain. In: Geiger CA (ed) Solid solutions in silicate and oxide systems, EMU notes in Mineralogy 3. Eötvös University Press, Budapest, pp 71–100

    Google Scholar 

  • Geiger CA (2008) Silicate garnet: a micro to macroscopic (re)view. Am Mineral 93:360–372

    Article  Google Scholar 

  • Haselton HT Jr, Westrum EF Jr (1980) Low-temperature heat capacities of synthetic pyrope, grossular, and pyrope60grossular40. Geochim Cosmochim Acta 44:701–709

    Article  Google Scholar 

  • Haselton HT Jr, Hovis GL, Hemingway BS, Robie RA (1983) Calorimetric investigation of the excess entropy of mixing in analbite-sanidine solid solutions: lack of evidence for Na, K short-range order and implications for two-feldspar thermometry. Am Mineral 86:398–413

    Google Scholar 

  • Holland TJB, Powell R (1998) An internally consistent thermodynamic data set for phases of petrological interest. J Metamorph Geol 16:309–343

    Article  Google Scholar 

  • Hovis GL (1988) Enthalpies and volumes related to K–Na mixing and Al–Si order/disorder in alkali feldspars. J Petrol 29:731–763

    Google Scholar 

  • Kroll H, Kotelnikov AR, Göttlicher J, Valyashko EV (1995) (K, Sr)-feldspar solid solutions: the volume behavior of heterovalent feldspars. Eur J Mineral 7:489–499

    Google Scholar 

  • Kubaschewski O, Alcock CB (1979) Metallurgical thermochemistry. Pergamon Press, Oxford

    Google Scholar 

  • Mao HK, Bell PM (1979) Equations of state of MgO and ε Fe under static pressure conditions. J Geophys Res 84:4533–4536

    Article  Google Scholar 

  • Oberti R, Quartieri S, Dalconi MC, Boscherini F, Iezzi G, Boiocchi M (2006) Distinct local environments for Ca along the non-ideal pyrope-grossular solid solution: a new model based on crystallographic and EXAFS analysis. Chem Geol 225:347–359

    Article  Google Scholar 

  • Oda H, Anderson OL, Isaak DG, Suzuki I (1992) Measurement of elastic properties of single-crystal CaO up to 1200 K. Phys Chem Miner 19:96–105

    Article  Google Scholar 

  • Quartieri S, Boscherini F, Dalconi C, Iezzi G, Meneghini C, Oberti R (2008) Magnesium K-edge EXAFS study of bond-length behavior in synthetic pyrope-grossular garnet solid solutions. Am Mineral 93:495–498

    Article  Google Scholar 

  • Schwab RG, Küstner D (1977) Präzisionsgitterkonstantenbestimmung zur Festlegung röntgenographischer Bestimmungskurven für synthetische Olivine der Mischkristallreihe Forsterit-Fayalit. Neues Jahrbuch Mineral Monatsh, 205–215

  • Shannon RD, Prewitt CT (1969) Effective ionic radii in oxides and fluorides. Acta Crystallogr B25:925–946

    Google Scholar 

  • Urusov VS (1992) A geometric model of deviations from Vegard’s rule. J Solid State Chem 98:223–236

    Article  Google Scholar 

  • Urusov VS (2001) Phenomenological theory of solid solutions. In: Geiger CA (ed) Solid solutions in silicate and oxide systems, EMU notes in Mineralogy 3. Eötvös University Press, Budapest, pp 121–153

    Google Scholar 

  • Urusov VS, Petrova TG, Leonenko EV, Eremin NN (2007) A computer simulation of halite-sylvite (NaCl–KCl) solid solution local structure, properties, and stability. Mosc Univ Geol Bull 62:117–122

    Article  Google Scholar 

  • Urusov VS, Petrova TG, Eremin NN (2008) Computer modeling of the local structure, mixing properties, and stability of solid solutions of alkaline-earth metal oxides. Crystallogr Rep 53:1030–1038

    Article  Google Scholar 

  • Van de Walle A, Ceder G (2002) The effect of lattice vibrations on substitutional alloy thermodynamics. Rev Mod Phys 74:11–45

    Article  Google Scholar 

  • Vinograd V, Winkler B (2010) An efficient cluster expansion method for binary solid solutions: application to the halite–silvite, NaCl–KCl, system. In: Rosso JJ (ed) Theoretical and computational methods in mineral physics: geophysical applications. Reviews in mineralogy and geochemistry, vol 71. American Mineralogical Society, Washington, DC, pp 413–436

    Google Scholar 

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Acknowledgments

This work was supported by a grant from the Austrian Science Fund (FWF), project number P 20210-N10, which is gratefully acknowledged. We thank M. Grodzicki, Salzburg, and C. A. Geiger, Salzburg, for their valuable contributions. The manuscript benefited much from the constructive review by V. Vinograd, Frankfurt, and an unknown reviewer.

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  1. Materialforschung und Physik, Universität Salzburg, Hellbrunnerstr. 34, 5020, Salzburg, Austria

    Artur Benisek & Edgar Dachs

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  1. Artur Benisek
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  2. Edgar Dachs
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Correspondence to Artur Benisek.

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Benisek, A., Dachs, E. On the nature of the excess heat capacity of mixing. Phys Chem Minerals 38, 185–191 (2011). https://doi.org/10.1007/s00269-010-0394-z

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