Artur Benisek

New developments in two-feldspar thermometry

Abstract The thermodynamic model of the two-feldspar thermometer has been revised. From recent enthalpy and volume measurements in the (Na,Ca)- and (K,Ca)-feldspar binaries, new interaction parameters have been derived and previous ones have been updated. Entropy parameters have been fitted to the phase equilibrium data of Seck (1971) and Elkins and Grove (1990). The two data sets could be suitably combined into one. Ideal Ab, Or, and An activities have been expressed in terms of both the molecular mixing and Al-avoidance models. Two-feldspar pairs from high-grade metamorphic rocks that cooled slowly under dry conditions suffer from a distinct type of retrograde resetting. Whereas the original An content in both the plagioclase and the alkali feldspar is preserved because the intercrystalline Ca + Al ↔ (Na,K) + Si diffusion is sluggish, Na and K may be freely exchanged between phases. Mathematical reversal of the Na-K exchange at constant An yields the temperature at which the two feldspars originally coexisted. The shifts in Ab and Or contents obtained from the reversal reflect the relative plagioclase/alkali feldspar proportions observed in thin sections. Good agreement between calculated and measured ratios was found for feldspar pairs from Sri Lankan granulites. This observation represents a successful test of the reliability of the calculated Ab-Or shifts. In contrast to dry metamorphic rocks, similar application of chemical constraints is not indicated in the case of volcanic rocks. Then the two-feldspar thermometer delivers three, usually incongruent temperatures: T(Ab), T(Or), and T(An). From the abundance of temperatures, Fuhrman and Lindsley (1988) suggested adjusting compositions within assumed chemical uncertainties (e.g., ±2 mol%) so that congruent temperatures could be obtained. However, the result is not unique. Depending on minute variations in the starting compositions, the temperatures may vary by several tens of degrees. In addition, temperatures vary to a similar extent depending on the type of search algorithm. Therefore, we advise users to completely abandon this practice. Instead, a statistical procedure is suggested: Two-feldspar compositions are randomly generated according to Gaussian distributions with their means at the observed compositions and standard errors chosen according to the quality of the chemical analysis. This procedure returns normally distributed temperatures [T(Ab), T(Or), T(An)] together with means and standard deviations. From the overlap of the three Gaussian curves the question of equilibrium or non-equilibrium crystallization of feldspar pairs may be addressed.

Excess heat capacity and entropy of mixing in high structural state plagioclase

Abstract Low- and high-temperature heat capacities for a series of synthetic high structural state plagioclase crystals (Ab-An) were measured using both a relaxation calorimeter and a differential scanning calorimeter. The measurements were performed at temperatures between 5 and 800 K on milligram-sized polycrystalline samples that had been characterized in a previous study. The data show positive excess heat capacities of mixing at temperatures below 300 K with a maximum value of ~2 J/(mol·K). Below ~70 K, the excess heat capacities exceed two standard deviations and are thus significant. Above 300 K, the measurements indicate negative excess heat capacities with a maximum of ca. -1.5 J/(mol·K) at about 400 K, and do not exceed two standard deviations. The excess vibrational entropies of mixing are positive with an asymmetric variation. At T = 298.15 K, the largest deviation from ideal behavior occurs at Ab20An80 amounting to ΔSexvib = 2.8 ± 2.4 J/(mol·K). An asymmetric Margules mixing model was found to adequately describe the vibrational entropy-composition behavior, yielding WSvibAbAn = 16.4 J/(mol·K) and = WSvibAnAb 4.7 J/(mol·K).

Excess heat capacity and entropy of mixing in ternary series of high-structural-state feldspars

Low- and high-temperature heat capacities were measured for ternary series of synthetic, high structural state feldspars in the NaAlSi 3 O 8 –KAlSi 3 O 8 –CaAl 2 Si 2 O 8 system that had been characterised by X-ray diffraction and transmission light microscopy. The compositions of the phases lie in the NaAlSi 3 O 8 -rich part of the system where the stability is less limited by the ternary miscibility gap than in other compositional fields. The heat capacities of the end-members had been measured in previous studies using the same calorimeters as in this study (relaxation calorimeter and differential scanning calorimeter). Below 300 K, all samples showed strong positive excess heat capacities of mixing giving rise to excess entropies of mixing up to Δ S ex = 5.8 ± 1.6 J mol −1 K −1 at T = 298.15 K. Above 300 K, further contributions to the excess entropies of mixing do not appear to be significant. The non-ideal entropic mixing behaviour was described by a ternary asymmetric Margules model, resulting in the ternary interaction parameter W AbOrAn S = 93.9 J mol −1 K −1 .

Almandine: Lattice and non-lattice heat capacity behavior and standard thermodynamic properties

Abstract The heat capacity of three synthetic polycrystalline almandine garnets (ideal formula Fe3Al2Si3O12) and one natural almandine-rich single crystal was measured. The samples were characterized by optical microscopy, electron microprobe analysis, X-ray powder diffraction, and Mössbauer spectroscopy. Measurements were performed in the temperature range 3 to 300 K using relaxation calorimetry and between 282 and 764 K using DSC methods. All garnets show a prominent λ-type heat-capacity anomaly at low temperatures resulting from a paramagnetic-antiferromagnetic phase transition. For two Fe3+-free or nearly Fe3+-free synthetic almandines, the phase transition is sharp and occurs at 9.2 K. Almandine samples that have ~3% Fe3+ show a λ-type peak that is less sharp and that occurs at 8.0 ± 0.2 K. The low-T CP data were adjusted slightly using the DSC results to improve the experimental accuracy. Integration of the low-T CP data yields calorimetric standard entropy, S°, values between 336.7 ± 0.8 and 337.8 ± 0.8 J/(mol·K). The smaller value is recommended as the best S° for end-member stoichiometric almandine, because it derives from the “best” Fe3+-free synthetic sample. The lattice (vibrational) heat capacity of almandine was calculated using the single-parameter phonon dispersion model of Komada and Westrum (1997), which allows the non-lattice heat capacity (Cex) behavior to be modeled. An analysis shows the presence of an electronic heat-capacity contribution (Cel, Schottky anomaly) superimposed on a larger magnetic heat-capacity effect (Cmag) around 17 K. The calculated lattice entropy at 298.15 K is Svib = 303.3 J/(mol·K) and it contributes about 90% to the total standard entropy at 298 K. The non-lattice entropy is Sex = 33.4 J/(mol·K) and consists of Smag = 32.1 J/(mol·K) and Sel = 1.3 J/(mol·K) contributions. The CP behavior for almandine above 298 K is given by the polynomial [in J/(mol·K)]: CP = 649.06(±4) - 3837.57(±122)⋅T-0.5 - 1.44682(±0.06)·107·T-2 + 1.94834(±0.09)·109·T-3 which is calculated using the measured DSC data together with one published heat-content datum determined by transposed-drop calorimetry along with a new determination in this work that gives H1181K - H302K = 415.0 ± 3.2 kJ/mol. Using our S° value and the CP polynomial for almandine, we derived the enthalpy of formation, ΔH°f, from an analysis of experimental phase equilibrium results on the reactions almandine + 3rutile = 3ilmenite + sillimanite + 2quartz and 2ilmenite = 2Fe + 2rutile + O2. A ΔH°f = -5269.63 kJ/mol was obtained.

Grossular: A crystal-chemical, calorimetric, and thermodynamic study

Abstract In spite of the amount of research that has been done on grossular, Ca3Al2Si3O12, there is still uncertainty regarding its exact thermodynamic properties. Because of insufficient sample characterization in the various published calorimetric studies, it is difficult to analyze conflicting CP and S○ results. To resolve the discrepancies, a detailed and systematic multi‑method investigation was undertaken. Three synthetic grossular samples and four natural grossular‑rich garnets were characterized by optical microscopy, electron microprobe analysis, IR, and MAS 29Si and 27Al NMR spectroscopy, and X-ray powder diffraction methods. Two of the natural grossulars, crystallized at relatively low temperatures, are optically anisotropic and two from the higher temperature amphibolite faces are isotropic. The natural garnets have between 94 and 97 mol% grossular with minor fractions of other garnet components, as well as small amounts of OH in solid solution. 29Si and 27Al MAS NMR spectra indicate that synthetic grossular crystallized at high‑P and high‑T conditions is ordered with respect to Al and Si. Heat-capacity measurements between 5 and 300 K were made using relaxation calorimetry and between 282 and 764 K using DSC methods. For the three synthetic grossulars, the CP data yield an average S○ value of 260.23 ± 2.10 J/(mol·K). The S○ values for the four natural grossular‑rich garnets, adjusted to end‑member grossular composition, range between 253.0 ± 1.2 and 255.2 ± 1.2 J/(mol·K). The results of this work thus confirm earlier low‑temperature adiabatic calorimetric studies that show small, but experimentally significant, differences in S° between natural and synthetic grossular samples. The difference in terms of heat-capacity behavior between synthetic and natural samples is that the latter have lower CP values at temperatures between 20 and 100 K by up to about 20%. Above 298 K, CP for grossular is given by CP J/(mol·K) = 556.18(±12) - 1289.97(±394)⋅T-0.5 - 2.44014(±0.24)⋅107⋅T-2 + 3.30386(±0.39)⋅109⋅T-3. Applying mathematical programming, published high‑P‑T results on the reaction 3anorthite = grossular + 2kyanite + quartz were analyzed thermodynamically. The calculations yield best‑fit values of ΔfH○ = -6627.0 kJ/mol and S○ = 258.8 J/(mol·K) for grossular. It is concluded that S○ ≈ 260 J/ (mol·K) is the best value for end‑member grossular. Variations in structural state and composition in natural samples, as well as assumptions used in correcting for solid‑solution and OH groups, appear to be the most important factors that could account for the smaller S○ values of 253-257 J/(mol·K).

Thermodynamic behavior and properties of katoite (hydrogrossular): A calorimetric study

Abstract The low-temperature heat capacity behavior of synthetic katoite, Ca3Al2H12O12, was investigated for the first time using microcalorimetry. The sample was synthesized hydrothermally in Au capsules at 250 °C and 3 kb water pressure. X-ray powder measurements show that about 98-99% katoite was obtained. Heat capacities were measured with a commercially designed relaxation calorimeter between 5 and 300 K on a milligram-sized sample and around ambient temperatures with a differential scanning calorimeter. The heat capacity data are well behaved at T < 300 K and show a monotonic decrease in magnitude with decreasing temperature. There is no evidence for any phase transition. A standard third-law entropy value of S° = 421.7 ± 1.6 J/(mol·K) was calculated. Published experimentally based S° values for katoite are slightly lower than this value. Estimations of S° based on empirical corresponding state schemes give S° values that are much too low. This is ultimately attributed to an inability to account for the vibration behavior of the OH groups in katoite that have very weak or no H-bonding. Using this new calorimetric-based S° value and published standard enthalpy of formation data for katoite, a calorimetric-based Gibbs free energy of formation at 298 K can be obtained as ΔG°f = -5021.2 ± 16.5 kJ/mol.

Experimentally Determined Standard Thermodynamic Properties of Synthetic MgSO4·4H2O (Starkeyite) and MgSO4·3H2O: A Revised Internally Consistent Thermodynamic Data Set for Magnesium Sulfate Hydrates

The enthalpies of formation of synthetic MgSO(4)·4H(2)O (starkeyite) and MgSO(4)·3H(2)O were obtained by solution calorimetry at T=298.15 K. The resulting enthalpies of formation from the elements are [Formula: see text] (starkeyite)=-2498.7±1.1 kJ·mol(-1) and [Formula: see text] (MgSO(4)·3H(2)O)=-2210.3±1.3 kJ·mol(-1). The standard entropy of starkeyite was derived from low-temperature heat capacity measurements acquired with a physical property measurement system (PPMS) in the temperature range 5 K<T<300 K: [Formula: see text] (starkeyite)=254.48±2.0 J·K(-1)·mol(-1). Additionally, differential scanning calorimetry (DSC) measurements with a Perkin Elmer Diamond DSC in the temperature range 270 K<T<300 K were performed to check the reproducibility of the PPMS measurements around ambient temperature. The experimental C(p) data of starkeyite between 229 and 303 K were fitted with a Maier-Kelley polynomial, yielding C(p)(T)=107.925+0.5532·T-1048894·T(-2). The hydration state of all Mg sulfate hydrates changes in response to local temperature and humidity conditions. Based on recently reported equilibrium relative humidities and the new standard properties described above, the internally consistent thermodynamic database for the MgSO(4)·nH(2)O system was refined by a mathematical programming (MAP) analysis. As can be seen from the resulting phase diagrams, starkeyite is metastable in the entire T-%RH range. Due to kinetic limitations of kieserite formation, metastable occurrence of starkeyite might be possible under martian conditions.

The heat capacity of fayalite at high temperatures

Abstract The high-temperature heat capacity of fayalite was reinvestigated using drop and differential scanning calorimetry. The resulting data together with drop calorimetry data taken from the literature were analyzed yielding CP J/(mol·K) = -584.388 + 129 440·T−1 - 3.84956·107·T−2 + 4.10143·109·T−3 + 98.4368·ln(T). This new CP polynomial is recommended for calculating phase equilibria involving fayalite at mantle conditions. Using thermal expansion coefficient and isothermal bulk modulus data from the literature, the isochoric heat capacity was calculated resulting in CV J/(mol·K) = - 217.137 + 63 023.1·T−1 - 2.15863·107·T−2 + 2.23513·109·T−3 + 51.7620·ln(T).

Thermochemistry of the alkali feldspars: Calorimetric study of the entropy relations in the low albite–low microcline series

Abstract New heat capacity data obtained on 12 samples of the low albite-low microcline series are presented. They were measured by relaxation and differential scanning calorimetry between 5 and 773 K. Two series, differing in their starting materials, were investigated, both of which were prepared via molten salt and solid-solid ion-exchange techniques in previous studies. The heat capacity of both series deviates positively from the ideal behavior leading to positive excess vibrational entropies of mixing, which can be described by a Margules mixing model yielding WSAbOr = 8.60 and WSOrAb = 9.28 J/(mol·K). The heat capacity and the vibrational entropy obtained on these Al,Si ordered samples are compared with those described in the literature for disordered samples. The solvi of the Al,Si ordered and disordered alkali feldspar systems were calculated from the calorimetric data and compared to experimentally determined solvi. Large deviations are detected for the ordered system, whereas consistent results are found for the disordered system, provided Na,K clustering is taken into account.

Heat capacity and third-law entropy of kaersutite, pargasite, fluoropargasite, tremolite and fluorotremolite

The heat capacities ( C p) of natural kaersutite, tremolite and fluoropargasite, as well as of synthetic pargasite were measured in the temperature range from 5 to 764 K and were used to calculate the standard entropy ( S °) of the pure end-members. The data between 5 and 300 K were obtained by relaxation calorimetry, the high-temperature data (282–764 K) by differential scanning calorimetry (DSC). Natural kaersutite was characterized by electron microprobe analysis and Mossbauer spectroscopy, the other samples had been characterized previously. Assuming ideal OH–F mixing, the C p data of natural fluoropargasite ( X F = 0.605) were used to estimate the heat capacity and standard entropy difference between OH and F end-members of pargasite and tremolite. Fits to the DSC data yielded the following polynomials for the corrected pure end-members (valid above 298 K, T [K], C p [J/mol·K]): \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \begin{eqnarray*}&&Kaersutite\ {-}\ NaCa\_{2}(Mg\_{4}\ Ti^{4+})\ (Si\_{6}Al\_{2})O\_{22}(OH)O\\&&\mathit{C}\_{p}\ =\ 1145.3014\ {-}\ 3356.9700\mathit{T}^{{-}0.5}\ {-}\ 4.3995\ {\times}\ 10^{7}\mathit{T}^{{-}2}\ +\ 5.8164\ {\times}\ 10^{9}\mathit{T}^{{-}3}\end{eqnarray*} \end{document} \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \begin{eqnarray*}&&Pargasite\ {-}\ NaCa\_{2}(Mg\_{4}Al)(Si\_{6}Al\_{2})O\_{22}(OH)\_{2}\\&&\mathit{C}_{p}\ =\ 1251.9495\ {-}\ 5934.2927\mathit{T}^{{-}0.5}\ {-}\ 3.6235\ {\times}\ 10^{7}\mathit{T}^{{-}2}\ +\ 4.8999\ {\times}\ 10^{9}\mathit{T}^{{-}3}\end{eqnarray*} \end{document} \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \begin{eqnarray*}&&Fluoropargasite\ {-}\ NaCa\_{2}(Mg\_{4}Al)(Si\_{6}Al\_{2})O\_{22}F\_{2}\\&&\mathit{C}_{p}\ =\ 1218.8568\ {-}\ 5923.9772\mathit{T}^{{-}0.5}\ {-}\ 2.9302\ {\times}\ 10^{7}\mathit{T}^{{-}2}\ +\ 3.4496\ {\times}\ 10^{9}\mathit{T}^{{-}3}\end{eqnarray*} \end{document} \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \begin{eqnarray*}&&Tremolite\ {-}\ Ca\_{2}Mg\_{5}Si\_{8}O\_{22}(OH)\_{2}\\&&\mathit{C}\_{p}\ =\ 1278.1996\ {-}\ 8114.5798\mathit{T}^{{-}0.5}\ {-}\ 2.3199\ {\times}\ 10^{7}\mathit{T}^{{-}2}\ +\ 2.8845\ {\times}\ 10^{9}\mathit{T}^{{-}3}\end{eqnarray*} \end{document} \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \begin{eqnarray*}&&Fluorotremolite\ {-}\ Ca\_{2}Mg\_{5}Si\_{8}O\_{22}F\_{2}\\&&\mathit{C}\_{p}\ =\ 1145.1069\ {-}\ 8104.2643\mathit{T}^{{-}0.5}\ {-}\ 1.6266\ {\times}\ 10^{7}\mathit{T}^{{-}2}\ +\ 1.4342\ {\times}\ 10^{9}\mathit{T}^{{-}3.}\end{eqnarray*} \end{document} A comparison with the various C p- T polynomials and additive estimation techniques suggested in the literature for T > 300 K indicates that most of these lie in a corridor within ±1 % of the values derived here. However, none of these individual alternatives works equally well over the whole temperature range 300–1000 K. Fits of the low-temperature C p’s to a combination of Debye, Einstein and Schottky functions yielded a standard entropy of 599.7 ± 0.8 J/mol·K for kaersutite, 591.0 ± 4.7 J/mol·K for pargasite and 550.1 ± 0.8 J/mol·K for tremolite (no site-configurational entropy contributions added). These S ° values of pargasite and tremolite are in excellent agreement with previous determinations from phase equilibrium experiments and low-temperature adiabatic calorimetry. The entropy difference between OH and F pargasite was found to be 5.3 J/mol·K and S ° of fluoropargasite is 585.7 ± 1.2 J/mol·K. Assuming the same difference for tremolite, S ° of fluorotremolite amounts to 544.8 ± 0.8 J/mol·K. Similar to F-phlogopite and F-apatite, fluoropargasite has a larger heat capacity at low temperatures compared to its OH analogue with a cross-over around 50 K and a possible reason for this behaviour is discussed. Based on the S ° values of pargasite, tremolite and their F analogues from this study, and on the results of OH–F partitioning experiments from the literature, a standard reaction enthalpy Δ H Ro = −5.5 ± 1.0 kJ/mol for the pargasite-phlogopite F–OH exchange, and a Δ H Ro = − 14.8 ± 2.5 kJ/mol for the tremolite-phlogopite F–OH exchange were calculated. Standard enthalpy of formation values for fluoropargasite and fluorotremolite can then be derived.