Low-temperature heat capacities of MgAl<Subscript>2</Subscript>O<Subscript>4</Subscript> and spinels of the MgCr<Subscript>2</Subscript>O<Subscript>4</Subscript>–MgAl<Subscript>2</Subscript>O<Subscript>4</Subscript> solid solution

We present results from low-temperature heat capacity measurements of spinels along the solid solution between MgAl₂O₄ and MgCr₂O₄. The data also include new low-temperature heat capacity measurements for MgAl₂O₄ spinel. Heat capacities were measured between 1.5 and 300 K, and thermochemical functions were derived from the results. No heat capacity anomaly was observed for MgAl₂O₄ spinel; however, we observe a low-temperature heat capacity anomaly for Cr-bearing spinels at temperatures below 15 K. From our data we calculate standard entropies (298.15 K) for Mg(Cr,Al)₂O₄ spinels. We suggest a standard entropy for MgAl₂O₄ of 80.9 ± 0.6 J mol⁻¹ K⁻¹. For the solid solution between MgAl₂O₄ and MgCr₂O₄, we observe a linear increase of the standard entropies from 80.9 J mol⁻¹ K⁻¹ for MgAl₂O₄ to 118.3 J mol⁻¹ K⁻¹ for MgCr₂O₄.     

Thermodynamic functions of eskolaite Cr<Subscript>2</Subscript>O<Subscript>3</Subscript>(c) at 0–1800 K

The heat capacity of eskolaite Cr₂O₃(c) was determined by adiabatic vacuum calorimetry at 11.99–355.83 K and by differential calorimetry at 320–480 K. Experimental data of the authors and data compiled from the literature were applied to calculate the heat capacity, entropy, and the enthalpy change of Cr₂O₃ within the temperature range of 0–1800 K. These functions have the following values at 298.15 K: C _p ⁰ (298.15) = 121.5 ± 0.2 J K⁻¹mol⁻¹, S ⁰(298.15) = 80.95 ± 0.14 J K⁻¹mol⁻¹, and H ⁰(298.15)-H ⁰(0) = 15.30±0.02 kJ mol⁻¹. Data were obtained on the transitions from the antiferromagnetic to paramagnetic states at 228–457 K; it was determined that this transition has the following parameters: Neel temperature T _N = 307 K, Δ _tr S = 6.11 ± 0.12 J K⁻¹mol⁻¹ and δ _tr H = 1.87 ± 0.04 kJ mol⁻¹.     

Heat capacity and thermodynamic properties of GdPO<Subscript>4</Subscript> in the temperature range 0–1600 K

The heat capacity of gadolinium orthophosphate (GdPO₄) measured in the temperature range 11.15–344.11 K by adiabatic calorimetry and available literature data were used to calculate its thermodynamic functions at 0–1600 K. At 298.15 K, these functions are as follows: C _p ⁰(298.15 K) = 101.85 ± 0.05 J K⁻¹ mol⁻¹, S ⁰(298.15 K) = 123.82 ± 0.18 J K⁻¹ mol⁻¹, H ⁰(298.15 K)–H ⁰(0) = 17.250 ± 0.012 kJ mol⁻¹, and Φ ⁰(298.15 K) = 65.97 ± 0.18 J K⁻¹ mol⁻¹ The calculated Gibbs free energy of formation from the elements of GdPO₄ is Δ _f G ⁰ (298.15 K) = −1844.3 ± 4.7 kJ mol⁻¹.     

Heat capacity and thermodynamic functions of pretulite ScPO<Subscript>4</Subscript>(c) at 0–1600 K

The heat capacity of synthetic pretulite ScPO₄(c) was measured by adiabatic calorimetry within a temperature range of 12.13–345.31 K, and the temperature dependence of the pretulite heat capacity at 0–1600 K was derived from experimental and literature data on H ⁰(T)-H ⁰(298.15 K) for Sc orthophosphate. This dependence was used to calculate the values of the following thermodynamic functions: entropy, enthalpy change, and reduced Gibbs energy. They have the following values at 298.15 K: C _p ⁰ (298.15 K) = 97.45 ± 0.06 J K⁻¹ mol⁻¹, S ⁰(298.15 K) = 84.82 ± 0.18 J K⁻¹ mol⁻¹, H ⁰(298.15 K)-H ⁰(0) = 14.934 ± 0.016 kJ mol⁻¹, and Φ ⁰(298.15 K) = 34.73 ± 0.19 J K⁻¹mol⁻¹. The enthalpy of formation Δ _f H ⁰(ScPO₄, 298.15 K) = − 1893.6 ± 8.4 kJ mol⁻¹.     

Low-temperature thermodynamic properties of disordered zeolites of the natrolite group

 The heat capacity of paranatrolite and tetranatrolite with a disordered distribution of Al and Si atoms has been measured in the temperature range of 6–309 K using the adiabatic calorimetry technique. The composition of the samples is represented with the formula (Na_1.90K_0.22Ca_0.06)[Al_2.24Si_2.76O₁₀]·nH₂O, where n=3.10 for paranatrolite and n=2.31 for tetranatrolite. For both zeolites, thermodynamic functions (vibrational entropy, enthalpy, and free energy function) have been calculated. At T=298.15 K, the values of the heat capacity and entropy are 425.1 ± 0.8 and 419.1 ±0.8 J K⁻¹ mol⁻¹ for paranatrolite and 381.0 ± 0.7 and 383.2 ± 0.7 J K⁻¹ mol⁻¹ for tetranatrolite. Thermodynamic functions for tetranatrolite and paranatrolite with compositions corrected for the amount of extraframework cations and water molecules have also been calculated. The calculation for tetranatrolite with two water molecules and two extraframework cations per formula yields: C _p (298.15)=359.1 J K⁻¹ mol⁻¹, S(298.15) −S(0)=362.8 J K⁻¹ mol⁻¹. Comparing these values with the literature data for the (Al,Si)-ordered natrolite, we can conclude that the order in tetrahedral atoms does not affect the heat capacity. The analysis of derivatives dC/dT for natrolite, paranatrolite, and tetranatrolite has indicated that the water- cations subsystem within the highly hydrated zeolite may become unstable at temperatures above 200 K. Received: 30 July 2001 / Accepted: 15 November 2001     

Heat capacity and thermodynamic functions of xenotime YPO<Subscript>4</Subscript>(c) at 0–1600 K

The heat capacity of xenotime YPO₄(c) was measured by adiabatic calorimetry at 4.78–348.07 K. Our experimental and literature data on H ⁰(T)-H ⁰(298.15 K) of Y orthophosphate were utilized to derive the C _p ⁰(T) function of xenotime at 0–1600 K, which was then used to calculate the values of thermodynamic functions: entropy, enthalpy change, and reduced Gibbs energy. These functions assume the following values at 298.15 K: C _p ⁰ (298.15 K) = 99.27 ± 0.02 J K⁻¹ mol⁻¹, S ⁰(298.15 K) = 93.86 ± 0.08 J K⁻¹ mol⁻¹, H ⁰(298.15 K) − H ⁰(0) = 15.944 ± 0.005 kJ mol⁻¹, Φ⁰(298.15 K) = 40.38 ± 0.08 J K⁻¹ mol⁻¹. The value of the free energy of formation Δ _f G ⁰(YPO₄, 298.15 K) is −1867.9 ± 1.7 kJ mol⁻¹.     

Excess heat capacity and entropy of mixing in the high-structural state (K,Ca)-feldspar binary

Low- and high-temperature heat capacities were measured for a series of synthetic high-structural state (K,Ca)-feldspars (Or–An) using both a relaxation and a differential scanning calorimeter. The data were collected at temperatures between 5 and 800 K on polycrystalline samples that had been synthesised and characterised in a previous study. Below T = 300 K, Or₉₀An₁₀, and Or₈₀An₂₀ showed excess heat capacities of mixing with maximum values of ~3 J mol⁻¹ K⁻¹. The other members of this binary (An > 20 mol%) had lower excess heat capacity values of up to ~1 J mol⁻¹ K⁻¹. Above T = 300 K, some compositions exhibited negative excess heat capacities of mixing (with maximum values of −2 J mol⁻¹ K⁻¹). The vibrational entropy at 298.15 K for Or₉₀An₁₀ and Or₈₀An₂₀ deviated strongly from the behaviour of a mechanical mixture, with excess entropy values of ~3.5 J mol⁻¹ K⁻¹. More An-rich members had only small excess vibrational entropies at T = 298.15 K. The difference in behaviour between members with An > 20 mol% and those with An ≤ 20 mol% is probably a consequence of the structural state of the (K,Ca)-feldspars, i.e., (K,Ca)-feldspars with An ≤ 20 mol% have monoclinic symmetry, whereas those with An > 20 mol% are triclinic. At T = 800 K, the vibrational entropy values were found to scatter around the values expected for a mechanical mixture and, thus, correspond to a quasi-ideal behaviour. The solvus for the (K,Ca)-feldspar binary was calculated based on the entropy data from this study in combination with enthalpy and volume of mixing data from a previous study.     

Heat capacity of γ-Fe2SiO4 between 5 and 303 K and derived thermodynamic properties

A multi-anvil device was used to synthesize 24 mg of pure γ-Fe₂SiO₄ crystals at 8.5 GPa and 1,273 K. The low-temperature heat capacity (C _p) of γ-Fe₂SiO₄ was measured between 5 and 303 K using the heat capacity option of a physical properties measurement system. The measured heat capacity data show a broad λ-transition at 11.8 K. The difference in the C _p between fayalite and γ-Fe₂SiO₄ is reduced as the temperature increases in the range of 50–300 K. The gap in C _p data between 300 and 350 K of γ-Fe₂SiO₄ is an impediment to calculation of a precise C _p equation above 298 K that can be used for phase equilibrium calculations at high temperatures and high pressures. The C _p and entropy of γ-Fe₂SiO₄ at standard temperature and pressure (S°₂₉₈) are 131.1 ± 0.6 and 140.2 ± 0.4 J mol⁻¹ K⁻¹, respectively. The Gibbs free energy at standard pressure and temperature (ΔG° _f,298) is calculated to be −1,369.3 ± 2.7 J mol⁻¹ based on the new entropy data. The phase boundary for the fayalite–γ-Fe₂SiO₄ transition at 298 K based on current thermodynamic data is located at 2.4 ± 0.6 GPa with a slope of 25.4 bars/K, consistent with extrapolated results of previous experimental studies.     

Heat capacity and phase equilibria of wadeite-type K2Si4O9

The low-temperature heat capacity (C _p) of Si-wadeite (K₂Si₄O₉) synthesized with a piston cylinder device was measured over the range of 5–303 K using the heat capacity option of a physical properties measurement system. The entropy of Si-wadeite at standard temperature and pressure calculated from the measured heat capacity data is 253.8 ± 0.6 J mol⁻¹ K⁻¹, which is considerably larger than some of the previous estimated values. The calculated phase transition boundaries in the system K₂O–Al₂O₃–SiO₂ are generally consistent with previous experimental results. Together with our calculated phase boundaries, seven multi-anvil experiments at 1,400 K and 6.0–7.7 GPa suggest that no equilibrium stability field of kalsilite + coesite intervenes between the stability field of sanidine and that of coesite + kyanite + Si-wadeite, in contrast to previous predictions. First-order approximations were undertaken to calculate the phase diagram in the system K₂Si₄O₉ at lower pressure and temperature. Large discrepancies were shown between the calculated diagram compared with previously published versions, suggesting that further experimental or/and calorimetric work is needed to better constrain the low-pressure phase relations of the K₂Si₄O₉ polymorphs. Electronic supplementary material The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

High-temperature heat capacity and thermodynamic properties for end-member titanite (CaTiSiO5)

 The heat capacity of end-member titanite and (CaTiSiO₅) glass has been measured in the range 328–938 K using differential scanning calorimetry. The data show a weak λ-shaped anomaly at 483 ± 5 K, presumably associated with the well-known low-pressure P2₁/a ⇆ A2/a transition, in good agreement with previous studies. A value of 0.196 ± 0.007 kJ mol⁻¹ for the enthalpy of the P2₁/a ⇆ A2/a transition was determined by integration of the area under the curve for a temperature interval of 438–528 K, bracketing the anomaly. The heat capacity data for end-member titanite and (CaTiSiO₅) glass can be reproduced within <1% using the derived empirical equations (temperature in K, pressure in bars):

Excess heat capacity and entropy of mixing along the chlorapatite–fluorapatite binary join

The heat capacity at constant pressure, C _p, of chlorapatite [Ca₅(PO₄)₃Cl – ClAp], and fluorapatite [Ca₅(PO₄)₃F – FAp], as well as of 12 compositions along the chlorapatite–fluorapatite join have been measured using relaxation calorimetry [heat capacity option of the physical properties measurement system (PPMS)] and differential scanning calorimetry (DSC) in the temperature range 5–764 K. The chlor-fluorapatites were synthesized at 1,375–1,220°C from Ca₃(PO₄)₂ using the CaF₂–CaCl₂ flux method. Most of the chlor-fluorapatite compositions could be measured directly as single crystals using the PPMS such that they were attached to the sample platform of the calorimeter by a crystal face. However, the crystals were too small for the crystal face to be polished. In such cases, where the sample coupling was not optimal, an empirical procedure was developed to smoothly connect the PPMS to the DSC heat capacities around ambient T. The heat capacity of the end-members above 298 K can be represented by the polynomials: C _p^ClAp = 613.21 − 2,313.90T ^−0.5 − 1.87964 × 10⁷ T ⁻² + 2.79925 × 10⁹ T ⁻³ and C _p^FAp = 681.24 − 4,621.73 × T ^−0.5 − 6.38134 × 10⁶ T ⁻² + 7.38088 × 10⁸ T ⁻³ (units, J mol⁻¹ K⁻¹). Their standard third-law entropy, derived from the low-temperature heat capacity measurements, is S° = 400.6 ± 1.6 J mol⁻¹ K⁻¹ for chlorapatite and S° = 383.2 ± 1.5 J mol⁻¹ K⁻¹ for fluorapatite. Positive excess heat capacities of mixing, ΔC _p^ex, occur in the chlorapatite–fluorapatite solid solution around 80 K (and to a lesser degree at 200 K) and are asymmetrically distributed over the join reaching a maximum of 1.3 ± 0.3 J mol⁻¹ K⁻¹ for F-rich compositions. They are significant at these conditions exceeding the 2σ-uncertainty of the data. The excess entropy of mixing, ΔS ^ex, at 298 K reaches positive values of 3–4 J mol⁻¹ K⁻¹ in the F-rich portion of the binary, is, however, not significantly different from zero across the join within its 2σ-uncertainty.     

Heat capacity,entropy and phase equilibria of stishovite

The low-temperature heat capacity (C _P) of stishovite (SiO₂) synthesized with a multi-anvil device was measured over the range of 5–303 K using the heat capacity option of a physical properties measurement system (PPMS) and around ambient temperature using a differential scanning calorimeter (DSC). The entropy of stishovite at standard temperature and pressure calculated from DSC-corrected PPMS data is 24.94 J mol⁻¹ K⁻¹, which is considerably smaller (by 2.86 J mol⁻¹ K⁻¹) than that determined from adiabatic calorimetry (Holm et al. in Geochimica et Cosmochimica Acta 31:2289–2307, 1967) and about 4% larger than the recently reported value (Akaogi et al. in Am Mineral 96:1325–1330, 2011). The coesite–stishovite phase transition boundary calculated using the newly determined entropy value of stishovite agrees reasonably well with the previous experimental results by Zhang et al. (Phys Chem Miner 23:1–10, 1996). The calculated phase boundary of kyanite decomposition reaction is most comparable with the experimental study by Irifune et al. (Earth Planet Sci Lett 77:245–256, 1995) at low temperatures around 1,400 K, and the calculated slope in this temperature range is mostly consistent with that determined by in situ X-ray diffraction experiments (Ono et al. in Am Mineral 92:1624–1629, 2007).     

The reaction MgCr2O4 + SiO2 = Cr2O3 + MgSiO3 and the free energy of formation of magnesiochromite (MgCr2O4)

Low-temperature heat capacity measurements for MgCr₂O₄ have only been performed down to 52 K, and the commonly quoted third-law entropy at 298 K (106 J K⁻¹ mol⁻¹) was obtained by empirical extrapolation of these measurements to 0 K without considering the magnetic or electronic ordering contributions to the entropy. Subsequent magnetic measurements at low temperature reveal that the Néel temperature, at which magnetic ordering of the Cr³⁺ ions in MgCr₂O₄ occurs, is at ∼15 K. Hence a substantial contribution to the entropy of MgCr₂O₄ has been missed. We have determined the position of the near-univariant reaction MgCr₂O₄+SiO₂=MgSiO₃+Cr₂O₃. The reaction, which has a small positive slope in P-T space, has been bracketed at 100 K intervals between 1273 and 1773 K by reversal experiments. The reaction is extremely sluggish, and lengthy run times with a flux (H₂O, BaO-B₂O₃ or K₂O-B₂O₃) are needed to produce tight reversal brackets. The results, combined with assessed thermodynamic data for Cr₂O₃, MgSiO₃ and SiO₂, give the entropy and enthalpy of formation of MgCr₂O₄ spinel. As expected, our experimental results are not in good agreement with the presently available thermodynamic data. We obtain Δ _f H ^∘ ₂₉₈=−1759.2±1.5 kJ mol⁻¹ and S ^∘ ₂₉₈=122.1±1.0 J K⁻¹ mol⁻¹ for MgCr₂O₄. This entropy is some 16 J K⁻¹ mol⁻¹ more than the calorimetrically determined value, and implies a value for the magnetic entropy of MgCr₂O₄ consistent with an effective spin quantum number (S') for Cr³⁺ of 1/2 rather than the theoretical 3/2, indicating, as in other spinels, spin quenching. Received: 9 May 1997 / Accepted: 28 July 1997     

Thermodynamic data of the high-pressure phase Mg5Al5Si6O21(OH)7 (Mg-sursassite)

 Calorimetric and P–V–T data for the high-pressure phase Mg₅Al₅Si₆O₂₁(OH)₇ (Mg-sursassite) have been obtained. The enthalpy of drop solution of three different samples was measured by high-temperature oxide melt calorimetry in two laboratories (UC Davis, California, and Ruhr University Bochum, Germany) using lead borate (2PbO·B₂O₃) at T=700 ^∘C as solvent. The resulting values were used to calculate the enthalpy of formation from different thermodynamic datasets; they range from −221.1 to −259.4 kJ mol⁻¹ (formation from the oxides) respectively −13892.2 to −13927.9 kJ mol⁻¹ (formation from the elements). The heat capacity of Mg₅Al₅Si₆O₂₁(OH)₇ has been measured from T=50 ^∘C to T=500 ^∘C by differential scanning calorimetry in step-scanning mode. A Berman and Brown (1985)-type four-term equation represents the heat capacity over the entire temperature range to within the experimental uncertainty: C _P (Mg-sursassite) =(1571.104 −10560.89×T ^−0.5−26217890.0 ×T ⁻²+1798861000.0×T ⁻³) J K⁻¹ mol⁻¹ (T in K). The P V T behaviour of Mg-sursassite has been determined under high pressures and high temperatures up to 8 GPa and 800 ^∘C using a MAX 80 cubic anvil high-pressure apparatus. The samples were mixed with Vaseline to ensure hydrostatic pressure-transmitting conditions, NaCl served as an internal standard for pressure calibration. By fitting a Birch-Murnaghan EOS to the data, the bulk modulus was determined as 116.0±1.3 GPa, (K ^′=4), V _T,0 =446.49 ³ exp[∫(0.33±0.05) × 10⁻⁴ + (0.65±0.85)×10⁻⁸ T dT], (K _T/T) _P  = −0.011± 0.004 GPa K⁻¹. The thermodynamic data obtained for Mg-sursassite are consistent with phase equilibrium data reported recently (Fockenberg 1998); the best agreement was obtained with Δ_f H ⁰ ₂₉₈ (Mg-sursassite) = −13901.33 kJ mol⁻¹, and S ⁰ ₂₉₈ (Mg-sursassite) = 614.61 J K⁻¹ mol⁻¹. Received: 21 September 2000 / Accepted: 26 February 2001     

Heat capacity,entropy, and phase equilibria of dmitryivanovite

The heat capacity (C _p ) of dmitryivanovite synthesized with a cubic press was measured in the temperature range of 5–664 K using the heat capacity option of a physical properties measurement system and a differential scanning calorimeter. The entropy of dmitryivanovite at standard temperature and pressure (STP) was calculated to be 110.1 ± 1.6 J mol⁻¹ K⁻¹ from the measured C _p data. With the help of new phase equilibrium experiments done at 1.5 GPa, the phase transition boundary between krotite and dmitryivanovite was best represented by the equation: P (GPa) = −2.1825 + 0.0025 T (K). From the temperature intercept of this phase boundary and other available thermodynamic data for krotite and dmitryivanovite, the enthalpy of formation and Gibbs free energy of formation of dmitryivanovite at STP were calculated to be −2326.7 ± 2.1 and −2,208.1 ± 2.1 kJ mol⁻¹, respectively. It is also inferred that dmitryivanovite is the stable CaAl₂O₄ phase at STP and has a wide stability field at high pressures whereas the stability field of krotite is located at high temperatures and relatively low pressures. This conclusion is consistent with natural occurrences (in Ca–Al-rich inclusions) of dmitryivanovite and krotite, where the former is interpreted as the shock metamorphic product of originally present krotite.     

Thermodynamic properties of nickel chromite (NiCr2O4) based on adiabatic calorimetry at low temperatures

 The hitherto unknown low-temperature heat capacity of nickel chromite (NiCr₂O₄) was measured between 8 and 381 K using adiabatic calorimetry, and some thermochemical functions [C_P(T), S(T), S°₂₉₈, H_(T)−H₍₀₎] were derived from the results. The standard entropy (S°₂₉₈=140.0 ± 0.3 J mol⁻¹ K⁻¹) for nickel chromite was calculated from the results. Our calorimetric measurements indicate three major anomalies in the heat-capacity curve at temperatures between 8 and 381 K. A short literature review indicates that two of these anomalies can be accounted for, whereas an anomaly peaking at 29 K has not been reported previously.     

Heat capacity and phase equilibria of hollandite polymorph of KAlSi3O8

The low-temperature heat capacity (C _p ) of KAlSi₃O₈ with a hollandite structure was measured over the range of 5–303 K with a physical properties measurement system. The standard entropy of KAlSi₃O₈ hollandite is 166.2±0.2 J mol⁻¹ K⁻¹, including an 18.7 J mol⁻¹ K⁻¹ contribution from the configurational entropy due to disorder of Al and Si in the octahedral sites. The entropy of K₂Si₄O₉ with a wadeite structure (Si-wadeite) was also estimated to facilitate calculation of phase equilibria in the system K₂O–Al₂O₃–SiO₂. The calculated phase equilibria obtained using Perple_x are in general agreement with experimental studies. Calculated phase relations in the system K₂O–Al₂O₃–SiO₂ confirm a substantial stability field for kyanite–stishovite/coesite–Si-wadeite intervening between KAlSi₃O₈ hollandite and sanidine. The upper stability of kyanite is bounded by the reaction kyanite (Al₂SiO₅) = corundum (Al₂O₃) + stishovite (SiO₂), which is located at 13–14 GPa for 1,100–1,400 K. The entropy and enthalpy of formation for K-cymrite (KAlSi₃O₈·H₂O) were modified to better fit global best-fit compilations of thermodynamic data and experimental studies. Thermodynamic calculations were undertaken on the reaction of K-cymrite to KAlSi₃O₈ hollandite + H₂O, which is located at 8.3–10.0 GPa for the temperature range 800–1,600 K, well inside the stability field of stishovite. The reaction of muscovite to KAlSi₃O₈ hollandite + corundum + H₂O is placed at 10.0–10.6 GPa for the temperature range 900–1,500 K, in reasonable agreement with some but not all experiments on this reaction.     

Low T heat capacity measurements and new entropy data for titanite (sphene): implications for thermobarometry of high-pressure rocks

The accepted standard state entropy of titanite (sphene) has been questioned in several recent studies, which suggested a revision from the literature value 129.3 ± 0.8 J/mol K to values in the range of 110–120 J/mol K. The heat capacity of titanite was therefore re-measured with a PPMS in the range 5 to 300 K and the standard entropy of titanite was calculated as 127.2 ± 0.2 J/mol K, much closer to the original data than the suggested revisions. Volume parameters for a modified Murgnahan equation of state: V _P,T  = V ₂₉₈° × [1 + a°(T − 298) − 20a°(T − 298)] × [1 – 4P/(K ₂₉₈ × (1 – 1.5 × 10⁻⁴ [T − 298]) + 4P)]^1/4 were fit to recent unit cell determinations at elevated pressures and temperatures, yielding the constants V ₂₉₈° = 5.568 J/bar, a° = 3.1 × 10⁻⁵ K⁻¹, and K = 1,100 kbar. The standard Gibbs free energy of formation of titanite, −2456.2 kJ/mol (∆H°_f = −2598.4 kJ/mol) was calculated from the new entropy and volume data combined with data from experimental reversals on the reaction, titanite + kyanite = anorthite + rutile. This value is 4–11 kJ/mol less negative than that obtained from experimental determinations of the enthalpy of formation, and it is slightly more negative than values given in internally consistent databases. The displacement of most calculated phase equilibria involving titanite is not large except for reactions with small ∆S. Re-calculated baric estimates for several metamorphic suites yield pressure differences on the order of 2 kbar in eclogites and 10 kbar for ultra-high pressure titanite-bearing assemblages.     

Structure of magnetite (Fe<Subscript>3</Subscript>O<Subscript>4</Subscript>) above the Curie temperature: a cation ordering study

A pristine magnetite (Fe₃O₄) specimen was studied by means of Neutron Powder Diffraction in the 273–1,073 K temperature range, in order to characterize its structural and magnetic behavior at high temperatures. An accurate analysis of the collected data allowed the understanding of the behavior of the main structural and magnetic features of magnetite as a function of temperature. The magnetic moments of both tetrahedral and octahedral sites were extracted by means of magnetic diffraction up to the Curie temperature (between 773 and 873 K). A change in the thermal expansion coefficient around the Curie temperature together with an increase in the oxygen coordinate value above 700 K can be observed, both features being the result of a change in the thermal expansion of the tetrahedral site. This anomaly is not related to the magnetic transition but can be explained with an intervened cation reordering, as magnetite gradually transforms from a disordered configuration into a partially ordered one. Based on a simple model which takes into account the cation-oxygen bond length, the degree of order as a function of temperature and consequently the enthalpy and entropy of the reordering process were determined. The refined values are ΔH⁰ = −23.2(1.7) kJ mol⁻¹ and ΔS⁰ = −16(2) J K⁻¹ mol⁻¹. These results are in perfect agreement with values reported in literature (Mack et al. in Solid State Ion 135(1–4):625–630, 2000; Wu and Mason in J Am Ceramic Soc 64(9):520–522, 1981).     

Thermodynamic mixing behavior of synthetic Ca-Tschermak–diopside pyroxene solid solutions: I. Volume and heat capacity of mixing

The unite cell parameters and heat capacities of a series of synthetic clinopyroxenes on the join Ca-Tschermak (CaTs)−diopside (Di) were measured using X-ray powder diffraction and calorimetric methods, respectively. The volume of mixing at 298 K shows a negative asymmetric deviation from ideality. A two-parameter Margules fit to the data yields W _CaTs−Di ^V = −0.29 ± 0.11 cm³ mol⁻¹ and W _Di−CaTs ^V = −1.14 ± 0.14 cm³ mol⁻¹. Heat capacities were determined between 5 and 923 K by heat-pulse at 5−302 K and differential-scanning calorimetry at 143−923 K. The precision of the low and high temperature C _p data is better than ±1%. Polynomials of the form C _p = a + bT ^−1/2 + cT ⁻² + dT ⁻³ were fitted to the C _p data in the temperature range between 250 and 925 K. Thermal entropy values [S ₂₉₈ − S ₀] and [S ₉₀₀ −S ₀] as well as enthalpies [H ₂₉₈ − H ₀] and [H ₉₀₀ − H ₀] were calculated for all members of the solid solution series. No significant deviation from ideal mixing behavior was observed.