A calorimetric investigation of spessartine: Vibrational and magnetic heat capacity

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Geochimica et Cosmochimica Acta 73 (2009) 3393–3409 www.elsevier.com/locate/gca

A calorimetric investigation of spessartine: Vibrational and magnetic heat capacity Edgar Dachs a,*, Charles A. Geiger b, Anthony C. Withers c, Eric J. Essene d a b

Fachbereich Materialforschung & Physik, Abteilung Mineralogie, Universita¨t Salzburg, Hellbrunnerstrasse 34, A-5020 Salzburg, Austria Institut fu¨r Geowissenschaften, Abteilung Mineralogie, Christian-Albrechts-Universita¨t Kiel, Olshausenstrasse 40, D-24098 Kiel, Germany c Department of Geology and Geophysics, University of Minnesota, Minneapolis, MN 55455, USA d Department of Geological Sciences, University of Michigan, Ann Arbor, MI 48109-1005, USA Received 19 November 2008; accepted in revised form 13 March 2009; available online 20 March 2009

Abstract The heat capacity (Cp) of two synthetic spessartine samples (Sps) was measured on 20–30 mg-size samples in the temperature range 2–864 K by relaxation calorimetry (RC) and differential scanning calorimetry (DSC). The polycrystalline spessartine samples were synthesized in two different laboratories at high pressures and temperatures from glass and oxide-mixture starting materials and characterized by X-ray powder diffraction and electron-microprobe analysis. The low-temperature heat capacity data show a prominent lambda transition with a peak at 6.2 K, which is interpreted to be the result of a paramagnetic–antiferromagnetic phase transition. The DSC data around ambient T agree excellently with the RC data and can be represented by the Cp polynomial for T > 250 K: 0:5  1:45  107  T 2 þ 1:82  109  T 3 : C Sps p ¼ 610  3060  T

Integration of the low temperature Cp data yields a calorimetric standard entropy for the two different samples of So = 334.6 ± 2.7 J/mol  K and 336.0 ± 2.7 J/mol  K. The preferred standard third-law entropy for spessartine is So = 335.3 ± 3.8 J/mol  K, which is the mean value from the two separate determinations. The lattice (vibrational) heat capacity of spessartine was calculated using the single-parameter phonon dispersion model of Komada and Westrum. The lattice entropy at 298.15 K is S 298:15 ¼ 297:7 J=mol  K, which represents 89% of the calorimetric entropy. The magnetic heat vib capacity and entropy of spessartine, Smag, at 298.15 K were also calculated. The Smag of the two samples is 38.7 and 37.4 J/ mol  K, which is 87% and 83% of the maximum possible magnetic entropy given by 3Rln6 = 44.7 J/mol  K. Published modare analyzed and compared to the experimental data. el-dependent lattice-dynamic calculations of S 298:15 vib Using the calorimetrically determined So and the Cp polynomial for spessartine, together with high P–T experimental phase-equilibrium data on Mn2+–Mg partitioning between garnet and olivine, allows calculation of the standard enthalpy of formation of spessartine. This gives DH of ;Sps ¼ 5693:6  1:4 kJ=mol, a value nearly 50 kJ more negative than some published values. The Gibbs free energy of spessartine was also calculated and gives DGof ;Sps ¼ 5364:3 kJ=mol at 298.15 K. The new standard entropy and enthalpy of formation values for spessartine lead to revised estimates for the enthalpies of formation of other Mn2+-silicates. Resulting DH of values for Mn-biotite, Mn-chlorite, Mn-cordierite, Mn-staurolite and Mn-chloritoid are 7–34 kJ more negative than their values listed in the thermodynamic database ‘‘THERMOCALC”. As an example, the new standard entropy and enthalpy of formation for spessartine have been applied to Mn–Fe partitioning between garnet and orthopyroxene from manganiferous iron formations. Excellent agreement between the predicted and observed distribution coefficient was obtained. Ó 2009 Elsevier Ltd. All rights reserved.

*

Corresponding author. E-mail address: [email protected] (E. Dachs).

0016-7037/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.gca.2009.03.011

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1. INTRODUCTION Garnet is an important rock-forming mineral, and knowledge of its thermodynamic properties is imperative in order to undertake various petrological, geochemical and geophysical investigations. Over the past five decades many thermodynamic and crystal-chemical studies on aluminosilicate garnets [E3Al2Si3O12 with E = Fe2+(almandine), Mn2+ (spessartine—Sps), Mg (pyrope), and Ca (grossular)] have been made and much understanding has been reached (e.g., Geiger, 1999, 2008). However, there are still gaps of knowledge concerning the various thermodynamic functions for even the garnet end-members. This is the case for the Mn2+ aluminosilicate garnet, spessartine. Its third-law entropy has not been determined through low-temperature heat capacity measurements, nor has its enthalpy of formation been determined calorimetrically. Geiger and Armbruster (1997) measured Cp data of spessartine between about 350 and 900 K using DSC methods. Published third-law entropy values for spessartine were obtained through modeling of experimental phase-equilibrium data (Helgeson et al., 1978; Holland and Powell, 1998), model phonon density of state calculations estimated through IR and Raman spectra (Ottonello et al., 1996; Chopelas, 2006) and through rigid-ion lattice-dynamic calculations (Grammacioli et al., 2002). There is poor to fair agreement among the different studies in terms of the various derived third-law entropy values. One major unknown in all these studies is the nature of possible low-temperature magnetic contributions to Cp and S. They cannot be calculated quantitatively from existing data in the literature and, thus, direct calorimetric study is essential for them to be determined. 1.1. Goals of this work In this investigation heat-capacity measurements were made between 2 and 865 K on two synthetic spessartine samples using relaxation calorimetry (Lashley et al., 2003; Dachs and Bertoldi, 2005; Dachs and Geiger, 2006; Dachs et al., 2007) and differential scanning calorimetry (DSC) methods (Bosenick et al., 1996). The data allowed a direct determination of the third-law entropy of spessartine garnet for the first time. The low-temperature Cp data showed evidence for a magnetic phase transition and, therefore, both magnetic and lattice contributions to the total heat capacity and third-law entropy were analyzed. A short analysis and critique on various model-dependent lattice-dynamic results on the Cp and S of spessartine were also made. In addition, using the measured standard third-law entropy of spessartine, DH of and DGof for spessartine were derived and thermodynamic properties of other Mn2+-silicates were calculated. 2. SAMPLES AND EXPERIMENTAL METHODS 2.1. Synthesis and characterization of spessartine samples Two synthetic spessartines were used for calorimetric study. The synthesis methods for the first sample (label

S100#20), which was crystallized dry from a homogeneous glass at high pressures and temperatures without a flux at the University of Kiel, are described in Rodehorst et al. ˚ at 290 K, as gi(2002). Its unit-cell dimension is 11.6173 A ven by Rodehorst et al. (2004), who also studied the samples’ IR spectrum and low temperature thermal expansion behavior. The second sample (label A627) was synthesized using a piston-cylinder device at the University of Minnesota, using the cell assembly and calibration described in Xirouchakis et al. (2001). Reagent grade MnO, Al2O3 and SiO2 were mixed in stoichiometric proportions of spessartine, ground under ethanol in an agate pestle and mortar for 1 h and sealed in a 3 mm diameter Pt capsule. Because the cell, owing to the graphite furnace, is somewhat reducing, oxidation of Mn2+ in the encapsulated sample is not expected during synthesis. The starting material was reacted at 3 GPa and 1400 °C for 48 h. Run products from the initial reaction step were reground and annealed at 3 GPa and 1400 °C for a further 144 h. The resulting sample comprised euhedral crystals that appeared inclusion free when viewed under an optical microscope. They gave an ao cell dimen˚ based on a least-squares refinement sion of 11.6128(5) A of a powder X-ray pattern collected in the range between 10 and 135° 2h, using NBS 640b Si as internal standard. Electron microprobe measurements were undertaken on both spessartine samples with a JEOL microprobe at Kiel University using WDS methods at 15 kV and 15 nA with a beam size of 1 lm. The standards were corundum for Al, wollastonite for Ca and Si, MgO for Mg, natural tephroite for Mn and fayalite for Fe. The correction program employed was CITZAF. For sample S100#20 a number of separate grains were characterized by X-ray maps for the elements Mn, Al and Si, as well as through a number of point analyses. For sample A627 a total of seven different grains were analyzed with a number of point measurements per grain. 2.2. Calorimetric measurements from 2 to 300 K The low-temperature heat capacity of the two samples was measured with a commercially designed calorimeter (heat capacity option of the Physical Properties Measurement System (PPMS), constructed by Quantum DesignÒ). The measurements were performed at temperatures between 2 and 300 K on the polycrystalline spessartine samples, S100#20 and A627, weighing 29.5 and 24.0 mg, respectively. The garnets were contained in hermetically sealed Al containers. Heat capacity was measured at 60 different temperatures and three times at each temperature on cooling from 300 to 2 K with a logarithmic spacing. It has been demonstrated that such data compare well to those measured upon heating (Dachs and Bertoldi, 2005). The precision associated with the PPMS heat capacity method, given as 100  rCp/Cp, where rCp is one standard deviation, using single-crystals or sintered powders is 0.3% at T > 50 K, increasing to 0.5% at T < 50 K (Dachs and Bertoldi, 2005). PPMS measurements on sample powders, sealed in Al-pans, show lower precisions of 0.5–0.7% at T > 100 K and 2–3% at T < 100 K. The accuracy of PPMS heat-capacity measurements for single-crystals is

Calorimetric investigation of spessartine

0.5 ± 0.8% in the temperature range 100 to 300 K, whereas measurements on sealed powders tend to be systematically lower by 1–2% compared to data from low-T adiabatic calorimetry. More details describing the experimental method, as well as the data acquisition and evaluation procedures, are given in Dachs and Bertoldi (2005) and are not repeated here. 2.3. Calorimetric measurements from 282 to 864 K Heat capacity above 282 K was measured with a Perkin Elmer Diamond DSC 7, recently set up at Salzburg University. The step-scanning mode, as developed and described by Bosenick et al. (1996), was employed. A heating rate of 10 K/min was used over a temperature interval of 100 K. A synthetic single-crystal of corundum (31.764 mg) served as the standard and its heat capacity values were taken from the National Bureau of Standards Certificate (Ditmars et al., 1982). The heat capacities of the blank run were subtracted from that of the standard and sample runs, respectively, using a MathematicaÒ program that followed the method described in Mraw (1988). The Cp measurements were repeated 4 to 5 times for each garnet sample. 3. RESULTS 3.1. Data evaluation The electron-microprobe results are summarized in Table 1. The analyses show that both spessartine samples are compositionally homogeneous and are stoichiometric within the error of the method. The number of Si and Al atoms in the garnet formula unit, based on 12 oxygens,

Table 1 Microprobe results of the two synthetic spessartines. Numbers in parentheses are one standard deviation and refer to the last digits. n = number of analyses, n.d. = not determined. Sample

S100#20 (n = 13)

A627 (n = 77)

wt.% oxides SiO2 TiO2 Al2O3 MnO FeO MgO CaO

37.49(37) n.d. 21.34(32) 43.15(49) 0.09(5) n.d. 0.02(2)

37.38(23) n.d. 21.35(15) 42.98(26) 0.10(5) 0.14(2) 0.01(1)

Total

102.09(52)

101.95(37)

Cations per 12 O atoms Si 3.012(31) Ti n.d. Al 2.021(26) Mn 2.936(31) Fe 0.006(3) Mg n.d. Ca 0.002(1)

3.006(13) n.d. 2.024(12) 2.928(17) 0.007(3) 0.016(3) 0.001(1)

Total

7.982(9)

7.977(20)

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are 3.0 and 2.0 within 2r uncertainty, respectively. The number of Mn cations is 2.94 and 2.93, which are a little lower than the stoichiometric value of 3.0. However, experience with microprobe analyses on fine-grained synthetic garnets indicates that this is likely an analytical artifact. The raw Cp data are given in the Appendix (for the DSC results, the mean and standard deviation for the repeated measurements are listed). An uncertainty of ±0.02 mg for the sample weight was adopted for converting the PPMS data from units of lJ  K1 to molar units of J  K1  mol1 (Dachs and Bertoldi, 2005). A heat-capacity polynomial of the general form Cp = k0 + k1  T0.5 + k2  T2 + k3  T3 + k4  T + k5  T2 + k6  T3 was used to fit the measured molar heat capacity data using the program MathematicaÒ. Because of the presence of a strong, sharp heat-capacity anomaly at low T, the Cp data were divided into several temperature intervals. The separately fitted intervals were joined by finding a common temperature where adjacent Cp intervals have the same heatcapacity value. The polynomial Cp = k0 + k1  T0.5 + k2  T2 + k3  T3 (Berman and Brown, 1985) was used for fitting the high-T data interval. The Cp below 2 K was estimated by a linear extrapolation to 0 K from a plot of Cp/T versus T2. The standard molar entropies, So, for both spessartines at 298.15 K were then calculated by solving analytically and stepwise the integral (assuming ST=0 K = 0): Z 298:15 CP dT : ð1Þ S o  S T ¼0 K ¼ T 0 The uncertainty in So was calculated to be ±2.7 J/mol  K. This value is 0.8% of the total entropy and results from a further analysis of the PPMS technique for polycrystalline and single-crystal samples (Benisek and Dachs, 2008). It includes not only the error in So resulting from an individual PPMS measurement (Dachs and Geiger, 2006, Eqs. (5)– (9)), but also takes into account uncertainties stemming from sample preparation. These become discernable when several PPMS data sets have been measured on different samples of the same material. It allows the reproducibility of the PPMS data to be determined. The So was also calculated by numerical integration of the Cp data using the MathematicaÒ functions ‘‘NIntegrate” for numerical integration, and ‘‘Interpolation” for linear interpolation between data points. The two calculated standard entropies agreed within 0.09%. The Cp values given in Table 2 are smoothed values based on the polynomial fits. 3.2. Heat capacity of spessartine Fig. 1 shows a plot of the PPMS and DSC heat capacity values for the two synthetic spessartines. No correction to the experimental Cp data was made in terms of composition for either sample. Heat capacity measurements down to 2 K (shown in the inset to Fig. 1) exhibit a strong k anomaly with Cp values reaching 55 J/mol  K. In almandine and andradite similar k anomalies occur at low T, and it was shown that Fe2+ and Fe3+ undergo antiferromagnetic ordering below their Ne´el temperatures of 7.5 and 11.5 K, respectively (Prandel, 1971; Robie et al., 1987; Anovitz

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E. Dachs et al. / Geochimica et Cosmochimica Acta 73 (2009) 3393–3409 Table 2 Heat capacity at 298.15 K and calorimetric standard third-law entropy, So, of spessartine from this study and various values from the literature. The lattice contribution to the standard third-law entropy, S ovib , is also given. Sample/Reference Calorimetry (this study) S100#20 A627 Preferred valuea

Fig. 1. Measured low and high temperature heat capacity for synthetic spessartine samples S100#20 (squares) and A627 (triangles). Note that the Cps of A627 are offset by +30 J/mol  K to show both data sets in one plot. Open symbols represent the PPMS measurements (the mean value of the three PPMS Cp determinations at each temperature is shown) and filled symbols the DSC data. The low-temperature magnetic phase transition region is show in the inset. Error bars represent 2r and are smaller than the symbol size for the PPMS Cp measurements.

et al., 1993). Because Mn2+ has a half-filled d-shell, as does Fe3+, it is likely that spessartine orders antiferromagnetically as well. If this is the case, the peak of the k anomaly at 6.2 K, which is identical for both synthetic spessartines, corresponds to the Ne´el temperature. Sharp k-transitions have been found in tephroite (Mn2SiO4) at 47 K (Robie et al., 1982) and rhodochrosite (MnCO3) at 34 K (Robie et al., 1984). Robie et al. (1995) found a strong antiferromagnetic–paramagnetic transition for braunite (Mn7SiO12) at 94 K, but their data on rhodonite (MnSiO3) showed only a weak and broad transition at ca. 10 K. They attributed the lack of a sharp k-transition in rhodonite to an irregularity in the coordination polyhedra. The measured standard molar heat capacity, C 298:15 , and p standard entropy, So, of spessartine are given in Table 2 and compared to various values from the literature. The Cp of spessartine A627 is generally somewhat larger than that of S100#20 with a maximum difference of about 4.5% around 30 K, decreasing to <1% around 300 K and to <0.5% at higher temperatures. The Cp difference, DC p ¼ C A627  C S100#20 , p p between the two samples, as a function of temperature (Fig. 2), is less than one standard deviation of the heat capacity data (rCp) for sample S100#20, and less than 2rCp for sample A627 (except between 400 and 600 K), which has smaller heat capacity uncertainties. Thus, both Cp data sets agree within their respective experimental uncertainties. There is good agreement between Cp data measured with the relaxation and DSC methods at T  300 K. Their respective Cp values agree within one standard deviation. At 303.1 K, for example, the PPMS Cp of sample S100#20 is 343.0 ± 2.3 J/mol  K compared to 340.2 ± 2.6 J/mol  K resulting from the DSC measurement. At 303.0 K the PPMS measurements give 345.7 ± 2.3 J/mol  K for spessartine A627 compared to 342.0 ± 1.5 J/mol  K from the DSC measurement. The good agreement between the two Cp data sets indicates that the higher temperature PPMS Cp data have no measurable systematic error.

So C 298:15 p (J/mol  K) (J/mol  K)

S ovib (J/mol  K)

337.8 ± 2.3 334.6 ± 2.7 338.9 ± 2.3 336.0 ± 2.7 338.4 ± 3.3 335.3 ± 3.8

295.8 299.6 297.7

Spectroscopy (Kieffer model) Chopelas (2006) 339.3b Ottonello et al. (1996) 331.0

337.1b 331.5

292.4 283.96

Rigid ion model Grammacioli et al. (2002)

361.3c

282.3

Thermodynamic data sets Helgeson et al. (1978) — Holland and Powell (1998) 340.3d

311.7 367.0e

—

Empirical estimation methods Holland (1989) van Hinsberg et al. (2005) 337.9d

337.8 ± 4.1d 293.1 346.8d 302.1

—

a

Mean value from samples S100#20 and A627. Derived from the Kieffer model using the vibrational spectroscopic data of spessartine given in Chopelas (2006, Table 1): So includes a maximum magnetic contribution of 3Rln6. c Includes a term for static subsite Mn2+ disorder and a maximum magnetic contribution. d Based on estimation methods (see cited reference); Smag = 3Rln6 = 44.69 J/mol  K. e Adopted from Vance and Holland (1993). Spessartine properties were derived from Fe–Mn exchange data between garnet and pyroxene (Bhattacharya et al., 1990). b

Fig. 2. Uncertainty in the heat capacity data (2rCp) obtained on spessartine samples S100#20 (squares) and A627 (triangles) as function of temperature. They are given in relation to the difference  C S100#20 (stars). DCp in Cp between the two samples, DC p ¼ C A627 p p values are smaller than 2rCp (except between 400 and 600 K for A627).

The average deviation of the PPMS Cp data from the polynomial fits at 50 < T < 300 K is 0.3 ± 0.2% for sample S100#20 and 0.5 ± 0.4% for sample A627. They increase to

Calorimetric investigation of spessartine

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a maximum of ± 3 to 4% around the strong magnetic transition at T = 6.2 K. Above T > 300 K the average deviation of the DSC heat capacity data from the polynomial fit is 0.2 ± 0.2% for both spessartines. A plot of the deviations is shown in Fig. 3. Combining the PPMS Cp data above 210 K with the DSC data, and using the Cp polynomial of Berman and Brown (1985), yields for samples S100#20 and A627: 0:5  1:16  107  T 2 þ 1:41  109  T 3 C Sps p ¼ 628  3690  T

ð2aÞ and 0:5  1:74  107  T 2 þ 2:22  109  T 3 C Sps p ¼ 592  2430  T

ð2bÞ respectively (covariance matrices of the fits are available upon request). Eq. (2b) yields slightly larger values of Cp, by a maximum of 0.5%, than Eq. (2a). A final Cp polynomial for spessartine may be obtained by taking the mean of Eqs. (2a) and (2b) giving: C Sps p

¼ 610  3060  T

0:5

7

 1:45  10  T

2

9

þ 1:82  10  T

3

:

ð3Þ Polynomial (3) supercedes the experimentally based one given by Geiger and Armbruster (1997, Eq. (1)), that covers a smaller temperature range, although the differences between them are small. It should be used for thermodynamic calculations involving spessartine. Based on the relation Cp = Cv + T  V  a2  KT and using data for the molar volume V, the thermal expansivity a and the bulk modulus KT from Geiger (1999), Cv calculated from Eq. (3) reaches 98% of the Dulong–Petit limit at 1600 K (i.e., 489.2 compared to 498.9 J/mol  K). Extrapolation to still higher temperatures may underestimate Cp somewhat, because this value decreases continuously to 93% at 3000 K. At this temperature Eq. (3) yields a value of 27.6 J/afu  K (afu = atoms per formula unit), which is well within the range of 28.3 ± 2.0 J/afu  K determined by Berman and Brown (1985) for 91 different minerals. Fig. 4 shows the difference between Cp values calculated with Eq. (3) and that of Geiger and Armbruster (1997), as well as those obtained from

Fig. 4. Deviation of spessartine’s heat capacity, calculated from various literature sources ðC lit p Þ, from the Cp values given by the polynomial in this study (C calc p , Eq. (3) of text). BB85: Berman and Brown (1985); GA 97: Geiger and Armbruster (1997); HP98: Holland and Powell (1998); vH05: van Hinsberg et al. (2005); C06: Kieffer model and vibrational data of Chopelas (2006).

empirical estimation methods and thermodynamic data sets (Berman and Brown, 1985; Holland and Powell, 1998; van Hinsberg et al., 2005) and from spectroscopic-based lattice dynamic calculations (Ottonello et al., 1996; Chopelas, 2006). The deviation in all cases is less than 2% in the temperature range 300–1000 K, except for the Cp values of Ottonello et al. (1996), which are 3–8% smaller (outside the scale of Fig. 4). The deviation is least for the model of Chopelas (2006) and Cp calculated with the empirical method of van Hinsberg et al. (2005). The Cp polynomial of Geiger and Armbruster (1997) yields Cp values that are lower by about 1%, well within the experimental precision associated with the DSC method (Bosenick et al., 1996). The estimated Cp values according to Berman and Brown (1985) are systematically lower by 1.7% above 400 K. The Cp polynomial in THERMOCALC (Holland and Powell, 1998) gives Cp values almost 1.5% higher compared to Eq. (3) at low temperatures (350 K) and 2.5% lower around 1000 K. The deviation increases continuously towards higher temperatures with a region of similar heat capacity values around 500 K. 3.3. Standard entropy of spessartine

Fig. 3. Deviation of heat capacity from polynomial fits, as described in the text ðC fit p Þ, compared to the experimental heat capacities ðC exp p Þ for spessartine samples S100#20 and A627.

The standard entropies, So, of both spessartine samples agree excellently with one another (Table 2) with values of 334.6 ± 2.7 J/mol  K for sample S100#20, and 336.0 ± 2.7 J/mol  K for sample A627. The difference of 1.4 J/mol  K is within the experimental uncertainty associated with PPMS measurements on powders, which is about 0.8% (Benisek and Dachs, 2008). The mean value of the two gives So = 335.3 ± 3.8 J/mol  K. Smoothed values for thermodynamic functions of spessartine, based on the mean Cp of the two samples S100#20 and A627, are given in Table 3a (2–298 K) and Table 3b (298–1000 K). The standard third-law entropies of spessartine calculated from spectroscopic data are similar to the calorimetrically determined value, only if a full magnetic contribution of 3Rln6 is added to the vibrational entropy (Table 2). Ottonello et al. (1996) gave a value for So of 331.5 J/

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Table 3a Smoothed molar thermodynamic functions of spessartine between 2 and 298.15 K (M = 495.03 g/mol). Values were obtained by taking the mean Cp of the two spessartine samples S100#20 and A627. U  (ST  S0)  (HT  H0)/T. T (K)

Cp (J/mol  K)

(ST  S0) (J/mol  K)

(HT  H0)/T (J/mol  K)

U (J/mol  K)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 20 25 30 35 40 45 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 298.15

5.897 13.390 21.641 31.202 48.013 17.908 12.400 10.864 9.482 8.408 7.737 7.229 6.664 6.162 5.586 6.760 9.572 13.701 18.85 24.79 31.33 45.67 61.08 77.05 93.27 109.52 125.63 141.49 157.02 172.15 186.82 201.00 214.64 227.72 240.22 252.10 263.36 273.97 283.71 292.40 300.96 309.29 317.35 325.10 332.53 338.35

1.963 5.790 10.732 16.481 23.551 28.413 30.334 31.701 32.773 33.622 34.323 34.922 35.438 35.879 37.541 38.881 40.339 42.110 44.27 46.82 49.77 56.73 64.91 74.11 84.12 94.79 105.98 117.60 129.54 141.73 154.11 166.62 179.22 191.86 204.51 217.14 229.71 242.21 254.61 266.87 278.98 290.95 302.77 314.45 325.99 335.29

1.472 4.210 7.501 11.189 15.829 18.010 17.550 16.888 16.215 15.551 14.925 14.353 13.824 13.329 11.437 10.355 9.969 10.194 10.94 12.15 13.73 17.84 22.91 28.67 34.95 41.60 48.50 55.59 62.80 70.07 77.37 84.66 91.90 99.09 106.19 113.19 120.08 126.83 133.45 139.89 146.16 152.28 158.24 164.06 169.75 174.28

0.491 1.580 3.230 5.292 7.721 10.403 12.784 14.813 16.558 18.072 19.398 20.569 21.613 22.550 26.104 28.526 30.370 31.916 33.32 34.68 36.03 38.88 42.00 45.43 49.17 53.19 57.48 62.00 66.74 71.66 76.74 81.97 87.32 92.77 98.32 103.95 109.64 115.38 121.16 126.98 132.82 138.67 144.53 150.39 156.25 161.01

mol  K, and from the mode frequencies given by Chopelas (2006, Table 1) and the resulting S 298:15 for spessartine, a vavib lue of So = 337.1 J/mol  K is calculated. The So calculated using the method of Holland (1989) gives 337.8 ± 4.1 J/ mol  K, which is in good agreement with the calorimetric value. The estimation method of van Hinsberg et al. (2005) gives So = 346.8 J/mol  K, 3.4% greater than our calorimetrically derived So. Considerable disagreement exists with the value of So = 367.0 J/mol  K of Holland and

Powell (1998) and used in THERMOCALC. This value is almost 10% larger than the measured value and was taken from Vance and Holland (1993), who derived thermodynamic properties for spessartine from Fe–Mn partitioning data from natural coexisting garnet–pyroxene pairs (Bhattacharya et al., 1990). 3.4. Lattice and magnetic heat capacity and entropy of spessartine The molar heat capacity, Cp, of a crystalline substance can consist of a number of contributions (Grimvall, 2001) and can be expressed as: C p ¼ C vib þ C mag þ C el þ C def þ . . . ;

ð4Þ

where Cvib is the molar vibrational or lattice contribution, Cmag the molar magnetic, Cel the molar electronic, and Cdef the molar defect contribution; additional contributions could arise from a structural phase transition. In the case of spessartine, where Mn2+ contains five unpaired d-electrons there are no spin-allowed d-orbital electronic transitions. In addition, defect concentrations should not be large for end-member spessartine. Hence, the last two terms in Eq. (4) can be ignored in terms of analyzing Cp behavior. Spessartine shows a strong Cp anomaly at low T that is likely related to a paramagnetic/antiferromagnetic phase transition. Thus, following Eq. (4), the data are analyzed in terms of lattice and magnetic contributions to the total experimental Cp. 3.4.1. Lattice heat capacity and entropy In order to calculate the lattice heat capacity and entropy of spessartine, an empirical model (Komada and Westrum, 1997—abb. as KW) was employed. Komada (1986) developed a single-parameter phonon-dispersion model and used it to analyze the low T heat capacity behavior of the iron-silicates deerite and grunerite to obtain their lattice Cp (Komada et al., 1995). The KW model was used by Dachs et al. (2007) to calculate the lattice heat capacity of Fe–Mg olivine solid solutions. A key feature of the KW model is that, analogous to Debye theory, a characteristic temperature, i.e., hKW, is introduced and the heat capacity is expressed as a function of this single parameter. It can be determined by fitting the model to experimental heat capacity data (Komada, 1986; Komada and Westrum, 1997; Dachs et al., 2007). An attractive feature of this model is the behavior of hKW as a function of temperature, because it allows for an analysis and treatment of magnetic and electronic transitions. At temperatures above the magnetic transition, hKW approaches a constant value and this can be used to determine the lattice heat capacity in the region of the magnetic anomaly. A drop-off in the value of hKW at low temperatures reflects the existence of non-lattice Cp contributions due to magnetic and/or electronic effects. The KW model requires the following crystallographic and mineral–chemical input data: (1) Volume of the primitive unit-cell, Vp. Spessartine has space group Ia-3d and the primitive cell has 1/2 the volume of the I-centered unit-cell. For the unit-cell constant of spessartine, the value ˚ was used (Rodehorst et al., 2004). (2) The ao = 11.6173 A

Calorimetric investigation of spessartine

3399

Table 3b Smoothed molar thermodynamic functions of spessartine between 298 and 1000 K. The DGof values (formation from the elements) were calculated from Cp, DH of and So for spessartine as derived in this study, combined with thermodynamic properties of the elements from Robie and Hemingway (1995). U  (ST  S0)  (HT  H298)/T. T (K)

Cp (J/mol  K)

(ST  S0) (J/mol  K)

(HT  H298)/T (J/mol  K)

U (J/mol  K)

DGof (kJ/mol)

298.15 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000

338.3 339.6 370.6 394.8 414.0 429.6 442.5 453.2 462.3 470.1 476.8 482.7 487.9 492.5 496.7 500.6

335.3 337.4 392.2 443.3 491.0 535.4 577.0 616.0 652.6 687.2 719.8 750.8 780.2 808.2 835.0 860.6

0.00 2.09 52.62 93.94 128.48 157.84 183.15 205.22 224.66 241.92 257.36 271.26 283.85 295.32 305.81 315.45

335.3 335.3 339.6 349.4 362.5 377.6 393.8 410.7 428.0 445.3 462.5 479.5 496.4 512.9 529.2 545.1

5364.25 5362.20 5306.90 5251.56 5196.30 5141.16 5086.16 5031.31 4976.61 4922.06 4867.65 4813.36 4759.20 4705.15 4646.50 4592.16

shortest interatomic distance, d, which in the spessartine ˚ (Geiger crystal structure is the Si–O distance of 1.640 A and Armbruster, 1997). (3) The ratio of the shortest and longest distance from the center to the boundary of the first Brillouin zone to the radius, r, of the same volume sphere, Bl and Bh. Such symmetry points with distances rl and rh, respectively, have the coordinates (p/ao, p/ao, 0) and (0, 2p/ao, 0) (Zak, 1969). Bl and Bh are then given by the ratios: Bl = rl/r, and Bh = rh/r, where r = (3Vrec/4p)(1/3) and Vrec is the primitive cell volume in reciprocal space. (4) Arithmetic mean of the molar masses for the group of the heaviest atoms in the formula unit, Mh. A mean for the masses of Mn, Al and Si (with factors 3, 2 and 3) was chosen for Mh. (5) Arithmetic mean of the molar masses for the group of the lightest atoms in the formula unit, Ml. The molar mass of oxygen was taken for Ml. (6) Ratio of the frequency of the longitudinal mode to that of the corresponding transverse mode, RLT. This ratio was calculated from the frequencies of the acoustic modes given in Chopelas (2006), which gives RLT = 1.703. The KW model calculates CV and, therefore, the small difference between Cp and CV is accounted for using the relationship Cp  CV = T  V  a2  KT with values for V, a, and KT from Geiger (1999). The difference between Cp and CV amounts to 0.7% of Cp at 298 K and becomes negligible below 100 K. Using the above data, we calculated hKW. A relatively temperature independent hKW value was obtained by introducing a factor that allows for the lower cut-off frequency of the optical branch to be adjusted. Heavy atoms in a crystal lower the lower frequency of the TO modes by a factor of (Mmean/Mh)1/2 (Komada and Westrum, 1997). We introduced a new parameter, which acts as a multiplier to this mass ratio (0.8 was used in the calculations). It allows for the lowest frequencies for the TO modes of the heavy atoms (e.g., Mn) to be adjustable. Fig. 5 shows a plot of hKW versus temperature for spessartine S100#20. Below 30 K there is a strong fall off in the value of hKW, indicating the contribution of the mag-

netic phase transition to the heat capacity. Above 50 K, hKW, according to the original KW model, is weakly temperature dependent following the relationship hKW = 139.06 + 0.033  T (square symbols in Fig. 5) and it increases by 3.3 K per 100 K. In order to improve the behavior of hKW, several modifications of the original model of Komada (1986) were tested. It was found that by adjusting the low frequency optical modes, as described above, one can improve the behavior of hKW such that it becomes nearly temperature independent, as given by hKW = 144.77 + 0.011  T (Fig. 5). Fig. 6 shows the resulting calculated lattice heat capacity for spessartine S100#20. Cvib obtained from the modified KW model approaches the measured heat capacities above 30 K. How well the Cvib models of spessartine represent the experimental Cp values at temperatures higher than the magnetic transition is displayed in Fig. 7. This figure shows that Cvib computed with the KW model (modified and original) is within ±1% of the experimental PPMS Cp’s above 60 K. Solving the integral

Fig. 5. Characteristic temperature, hKW, for the original (squares) and modified (dots) Komada–Westrum model (Komada and Westrum, 1997) applied to the heat capacity of spessartine S100#20. Above ca. 30 K, hKW approaches a nearly constant value given by hKW = 144.77 + 0.011  T.

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Fig. 6. Vibrational molar heat capacity, Cvib, of spessartine S100#20 calculated with the modified Komada–Westrum model (curve labeled KW, based on the original model—Komada and Westrum, 1997—is the dashed line close to it) and with the Kieffer model and vibrational data from Chopelas (2006, curve labeled calculated from the KW model C06). Above 30 K, C vib p approaches the experimental heat capacity values. The difference between Cp–Cv has been accounted for as described in the text.

Fig. 8. Magnetic molar heat capacity, Cmag, of spessartine S100#20 as a function of temperature. Values of Cmag were calculated by subtracting the vibrational heat capacity from the measured heat capacity. Cmag was recalculated (line) using Eqs. (6a) and (6b) in the text and with the parameters above and below the Ne´el temperature as given in Table 4.

Table 4 Values for the fit parameters used in Eqs. (6a) and (6b). The resulting calculated magnetic entropy, Smag, is also given. Numbers in parentheses represent one standard deviation and refer to the last digits. TN: Ne´el temperature. A corresponding plot for sample S100#20 is shown in Fig. 8.

Fig. 7. Difference between experimental Cp and lattice Cp of spessartine, as function of temperature, expressed as 100ðC exp p  C vib Þ=C vib . Cvib was calculated with the Komada– Westrum model (modified and original—Komada and Westrum, 1997—squares) and with the Kieffer model using spectroscopic data of Chopelas (2006, triangles).

S ovib  S T ¼0 ¼

Z 0

298:15

C vib P dT T

ð5Þ

with the modified KW model gives the following values for the vibrational entropy of spessartine at 298.15 K: for sample S100#20 we obtain S ovib ¼ 295:8 J=mol  K and for sample A627 S ovib ¼ 299:6 J=mol  K. Their mean value is 297.7 J/mol  K (Table 2). This amounts to 88.4% and 89.2% of the total standard entropy. Various published model S ovib values are also listed in Table 2. 3.4.2. Magnetic heat capacity and entropy To obtain the magnetic contribution, Cmag, to the calorimetric heat capacity, the lattice part, Cvib, was subtracted from the total calorimetric Cp values by calculating Cvib with the KW model, i.e., Cmag = Cp  Cvib (Fig. 8). For describing Cmag behavior around the sharp k-transition, we used, following Dachs et al. (2007), the equations (e.g., Grønvold and Sveen, 1974):

Parameter

S100#20

A627

TN (K)

6.2

6.2

T < TN A0 (J/K  mol) a0 B0 (J/K  mol)

34.92(20) 0.5581(43) 5.80(7)

35.33(32) 0.5880(75) 6.72(12)

T > TN A (J/K  mol) a B (J/K  mol) C (J/K2  mol) D (J/K1.5  mol)

3.42(72) 0.324(7) 10.3(7) 0.085(31) 0.58(32)

2.52(7) 0.488(10) 15.0(7) 0.057(32) 2.41(32)

Smag (J/K  mol)

38.7

37.4

" # a0 A0 jT  T N j  1 þ B0 for T < T N ; and ð6aÞ a0 TN a   pffiffiffiffi A jT  T N j  1 þ B þ CT þ D T for T > T N : ð6bÞ C mag ¼ a TN C mag ¼

The last two terms in Eq. (6b) were added to improve the fit of the high-temperature tail in C mag . The C mag data for spessartine can be satisfactorily fit with both equations, as shown in Fig. 8 (fitting parameters in Table 4). The magRnetic entropy, Smag, is determined by evaluating the integral ðC mag =T ÞdT over the temperature interval shown in Fig. 8. It amounts to 38.7 and 37.4 J/mol  K for samples S100#20 and A627, respectively. This is 86.6% and 83.3% of the maximum possible magnetic entropy that is given by 3Rln6 = 44.7 J/mol  K. The sum of Cvib + Cmag is plotted in Fig. 9 for spessartine S100#20, together with the experimental PPMS Cp data. The maximum deviation reaches about 10% in the

Calorimetric investigation of spessartine

Fig. 9. Vibrational and magnetic molar heat capacity of spessartine S100#20 as a function of temperature. Cvib was computed with the modified Komada–Westrum model and C mag with Eqs. (6a), (6b) and parameters given in Table 4. The sum Cvib + Cmag describes satisfactorily the experimental heat capacity of spessartine (squares).

vicinity of TN, but it decreases to a few percent below and above TN. The sum S mag þ S ovib , derived from modeling Cvib and Cmag for samples S100#20 and A627, amounts to 334.5 and 337.0 J/mol  K, respectively. They are very close to the values calculated from the Cp polynomial fits (Table 2) that and So = 336.0 ± give So = 334.6 ± 2.7 J/mol  K 2.7 J/mol  K. 4. DISCUSSION

3401

with experiment. Based on the model adps, the vibrational entropy for spessartine at 298 K was calculated as 282.3 J/ mol  K (Table 2). Chopelas (2006) applied a simplified phonon density of state model from Kieffer (e.g., Kieffer, 1979, 1985), using primarily IR and Raman results on spessartine, to calculate Cvib. We repeated these Kieffer-model calculations using the data in Chopelas (2006). The results of this exercise, obtained for the temperature range 0 to 100 K, are shown in Fig. 6. The model Cvib values are less than those given by experiment. They are 1–2% higher at T > 100 K. The resulting S ovib value at 298 K obtained from the model Cvib values is 292.4 J/mol  K (Table 2). Both model S ovib values underestimate the experimentally based value of S ovib ¼ 297:7 J=mol  K, as does that of Ottonello et al. (1996), who obtained S ovib ¼ 283:96 J=mol  K. The model calculations underestimate Cvib at low temperatures and, thus, the vibrational entropy of spessartine. The difficulty may be due to the problem in describing quantitatively the large amplitude anisotropic vibration of the Mn2+ cations (Geiger and Armbruster, 1997) and their expression in the vibrational spectra of spessartine. The issue is important, because one can conclude that there is no justification to include, ad hoc, configurational entropy terms from the E-cation site to obtain the third-law entropy of spessartine (Grammacioli et al., 2002; Chopelas, 2006). Moreover, models that include a full magnetic entropy term to predict So, as in the case of spessartine where 3Rln6 = 44.52 J/ mole  K, must also be carefully considered in light of the experimental Cp data.

4.1. An analysis of model Cp calculations on spessartine

4.2. Standard enthalpy of formation of spessartine

Over the past three decades various lattice dynamic models have been used to determine mineral properties such as thermodynamic functions. Silicate garnet has received special attention in these studies (e.g., Ottonello et al., 1996; Mittal et al., 2001; Grammacioli et al., 2002; Chopelas, 2006), because the extensive experimental crystal-chemical, thermodynamic and vibrational spectroscopic results on them allow the models to be constrained as well as tested. The general lattice dynamic problem is not simple, because the E-cation in aluminosilicate garnet is characterized by large amplitude anisotropic vibration (i.e., ‘‘rattling or dynamic disorder”), especially in the case of Mg, Fe2+ and Mn2+ (e.g., Geiger et al., 1992; Geiger and Armbruster, 1997; Sani et al., 2004). The issue is central to the model calculations, because lattice vibrations involving these cations express themselves in the low-temperature Cp region. We analyze here the results from the lattice dynamic type investigations of Grammacioli et al. (2002) and Chopelas (2006). Grammacioli et al. (2002) calculated the vibrational heat capacity and entropy for spessartine using a rigid-ion lattice-dynamic model. Their work placed special emphasis on obtaining correct values for the atomic displacement parameters (adps) of the atoms in the structure. Their model adps, as expressed by Beq, values for Mn2+ underestimate the experimentally measured ones (Geiger and Armbruster, 1997) by roughly 15%. In comparison, model Beq values for Al, Si, and O are in better agreement

The calorimetrically determined standard entropy of spessartine was used together with experimental phaseequilibrium data of Wood et al. (1994) on Mn–Mg partitioning between garnet and olivine to derive the standard enthalpy of formation for spessartine. The Mg–Mn exchange reaction is: MgAl2=3 SiO4 þ MnSi1=2 O2 ¼ MnAl2=3 SiO4 þ MgSi1=2 O2 ð7Þ pyrope þ tephroite ¼ spessartine þ forsterite ; where pyrope* (prp) and spessartine* (sps) are garnet components and tephroite* (tph) and forsterite* (fo) are olivine components written on a one-atom basis. The equilibrium constant is given by K¼

Ol Ol Grt Ol aGrt X Grt sps afo Mn X Mg cMn cMg ¼ ¼ K DK c; Ol Grt Ol Ol Grt aprp atph X Mg X Mn cGrt Mg cMn

ð8Þ

which may be rewritten as: RT ln K þ RT ln

cGrt cOl Mg Mg ¼ RT ln K þ RT ln : D cGrt cOl Mn Mn

ð9Þ

If we apply symmetrical solution models to describe the Mn–Mg mixing behavior in the olivine and garnet solid solutions, then RT ln and

cOl Mg Ol ¼ W Ol G ð2X Mn  1Þ; cOl Mn

ð10Þ

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RT ln

cGrt Mg Grt ¼ W Grt G ð2X Mn  1Þ: cGrt Mn

ð11Þ

Inserting into Eq. (9) leads to: Grt Ol Ol RT ln K þ W Grt G ð2X Mn  1Þ ¼ RT ln K D þ W G ð2X Mn  1Þ:

ð12Þ

Using the experimental KD values and adopting W Ol G ¼ 5:5  2:1 kJ=Mn-atom for the 1300 °C, and a 1.59 kJ smaller value for the 1000 °C experiments (Wood et al., 1994), the right-hand side of Eq. (12) (RHSEq. (12)) is known as function of P and T. Because RT lnK ¼ ðDGoR ÞP;T , Eq. (12) can be written as: Z T DCpR dT  DH oR  298:15   Z T ðDCpR =T ÞdT þ T DS oR þ 

298:15

Z

P

DV

o R dP

Eq: Grt þ W Grt G ð2X Mn  1Þ ¼ RHS

ð12Þ

;

ð13Þ

1

where DH oR , DS oR , and DV oR are the standard reaction enthalpy, entropy and volume for equilibrium (7), respectively. Collecting known terms and because DH oR ¼ DH of ;prp þ DH of ;tph  DH of ;sps  DH of ;fo leads to: Eq: Eq: ð12Þ Grt  DH of ;sps þ W Grt G ð2X Mn  1Þ ¼ RHS   Z T Z T þ DCpR dT  T DS oR þ ðDCpR =T ÞdT

þ

Z

298:15 P

298:15

DV oR dP  DH of ;prp  DH of ;tph þ DH of ;fo :

ð14Þ

1

Eq. (14) 10. Plot ) versus Fig.Grt  of the right-hand side of Eq. (14) (RHS 2X Mn  1 based on the experimental results of Wood et al. (1994) on Mn–Mg partitioning between garnet and olivine. A linear fit to the data is shown by the solid line and the dashed lines show the ±2r uncertainty envelope. The intercept corresponds to –(1/3) of the standard enthalpy of formation of spessartine.

A plot of the right-hand side of Eq. (14) (RHSEq. (14)) vs. Grt ð2X Grt Mn  1Þ should yield a straight line with W G given by o the slope and DH f ;sps as the intercept. For evaluating RHSEq. (14), Eq. (3) was used for the Cp polynomial, and So = 335.3 ± 3.8 J/mol  K as the standard entropy of spessartine. All other values were taken from Holland and Powell (1998, HP98). Fig. 10 shows the results of this exercise and a straight-line fit yields DH of ;sps ¼ 1897:9 0:5 kJ=mol and W Grt G ¼ 0:5  1:1 kJ=mol for 1-atom per formula unit. The standard enthalpy of formation of spessartine is then DH of ;Sps ¼ 5693:6  1:4 kJ=mol. The derived thermodynamic properties of spessartine are compiled in Table 5. As discussed by Wood et al. (1994), the range of W Ol G values that can be applied to their garnet–olivine partitioning data is 4.74–8.84 kJ (1300 °C experiments) and 3.15 to 7.25 kJ o (1000 °C experiments). With the larger values for W Ol G , DH f changes by 5 kJ to DH of ¼ 5688:2 kJ=mol and with lower values to DH of ¼ 5694:4 kJ=mol. The derived W Grt G ¼ 0:46 1:08 kJ=mol (1-atom per formula unit) is indistinguishable from zero. This indicates that Mn–Mg mixing should be ideal in garnet, which is in agreement with the results of Wood et al. (1994) who obtained W Grt G ¼ 1:5 2:5 kJ=mol. As discussed by Wood et al. (1994), Mg–Mn partitioning between garnet and olivine could not be as tightly constrained in their experiments at 1000 °C as at 1300 °C. If only the latter experiments were used to derive DH of of spessartine, this changes only slightly by 0.2 kJ to DH of ;Sps ¼ 5693:8  1:4 kJ=mol. The standard enthalpy of formation for spessartine is similar to the values proposed by Karpov and Kashik (1968) and Boeglin (1981), which are 5686.934 and 5686.66 kJ/mol, respectively. It is, however, 47.1 kJ/mol more negative than the value of 5646.4 kJ/mol given by HP98 using the same experiments and a standard entropy value of 367.0 J/mol  K for spessartine. The difference in the standard enthalpy of formation is partly caused by the T DS oR term in Eq. (14) (DS oR ¼ 7:4 J=mol  K resulting from S oSps of this study, compared to DS oR ¼ 3:2 J=mol  K using the updated HP98 data set), but also by the difference in DCpR, because our value is 3% lower than the heat capacity of spessartine at 1300 °C according to HP98. Recalculated KD values (Eq. (12)) using the thermodynamic data given in Table 5 and all other properties as mentioned above, are listed in Table 6 and compared to the experimentally determined data from Wood et al. (1994). In most cases, they agree with the experimental value within the uncertainty cited by the authors. Not surprisingly, similar KD values are calculated with the original spessartine properties from HP98, showing that both sets of properties describe the experimental Mn–Mg partitioning data equally well. Considerable differences, however, arise when other

Table 5 Molar thermodynamic properties for spessartine, Mn3Al2Si3O12, where DHof, So and Vo are the enthalpy of formation from the elements, the third-law entropy and the volume at standard conditions of Po ( 1 bar) and To ( 298.15 K). So was obtained from the low-temperature PPMS heat capacity measurements, the Cp function from the DSC measurements (Eq. (3) of text), and DH of from the phase-equilibrium data of Wood et al. (1994) on Mn–Mg partitioning between garnet and olivine. Vo was taken from Geiger (1999). DH of (kJ/mol)

So (J/K  mol)

Vo (J/bar  mol)

ko (J/K  mol)

k1 (J/K1/2  mol)

k2 ( 107) (J  K/mol)

k3 ( 109) (J  K2/mol)

5693.6 ± 1.4

335.3 ± 3.8

11.796 ± 0.003

610

3060

1.45

1.82

Cp = ko + k1  T0.5 + k2  T2 + k3  T3 (valid for T > 250 K).

Calorimetric investigation of spessartine Table 6 KD values for Mn–Mg partitioning between garnet and olivine rec ðK exp D Þ from Wood et al. (1994) and their recalculated values ðK D Þ using the thermodynamic properties of spessartine from this study (Table 5). All other thermodynamic values are from the updated Holland and Powell (1998) data set. The 1000 °C experiments were made at a pressure of 9 kbar, the 1300 °C experiments at 27 kbar. T (°C)

K exp D

K rec D

1000 1000 1000 1300 1300 1300 1300 1300 1300 1300 1300 1300 1300 1300 1300

4.46(0.55) 3.25(0.35) 2.93(0.8) 2.15(0.35) 3.61(0.32) 3.65(0.3) 3.33(0.14) 2.42(0.18) 3.48(0.14) 4.01(0.16) 2.52(0.16) 2.46(0.10) 3.44(0.3) 3.32(0.2) 3.81(1.00)

4.17 3.90 3.02 1.91 3.78 3.58 3.16 2.24 3.32 3.94 2.67 2.56 3.70 3.07 4.20

phase equilibria are computed. This is shown below for the case of Mn–Fe partitioning between garnet and orthopyroxene. In Table 3b, values for DGof , of spessartine are also listed. They were calculated using the Cp, DH of and So values for spessartine as derived in this study along with the thermodynamic properties for the elements as taken from Robie and Hemingway (1995).

3403

treatment of their KD data, corrected to the conditions of 4.5 kbar and 550 °C, and adopting the mixing models listed in Appendix 3 from Mahar et al. (1997), they obtained DGR values for Mn–Mg exchange equilibria between garnet-biotite, chlorite-biotite, cordierite-biotite, staurolite-biotite and chloritoid-biotite (Table 7). These values were used in THERMOCALC to generate enthalpy of formation values for the various Mn2+-silicates (Table 7). Such a thermodynamic analysis is possible if DH of of one end-member silicate is independently known and then used as an ‘‘anchor phase” for the calculation of DH of for the other phases. Mahar et al. (1997) adopted the DH of of spessartine as the ‘‘anchor phase”, whose value was derived from the experiments of Wood et al. (1994) using a standard entropy So = 367.0 J/mol  K (Vance and Holland, 1993). Based on the calorimetric results herein, this requires a reevaluation of the DH of values for various Mn2+-silicates. Thus, the DH of values for Mn2+-biotite (Mn-Bt), Mn2+chlorite (Mn-Chl), Mn2+-cordierite (Mn-Crd), Mn2+-staurolite (Mn-St) and Mn2+-chloritoid (Mn-Ctd) were recalculated from the exchange equilibria. This was done using the program PET (Dachs, 2004) and the calorimetrically derived So = 335.3 J/mol  K and DH of ¼ 5693:6 kJ=mol values for spessartine. All other properties are taken from an updated version of the original Holland and Powell (1998) data set (file th32.pd). The results of this excercise are given in Table 7. They show that DH of values for the various Mn2+-silicates become more negative by 7–34 kJ compared to those in THERMOCALC. The effect on phase relations in the KMnFMASHO system is beyond the scope of this contribution and will be given in a forthcoming publication.

4.3. Thermodynamic properties of other Mn2+- silicates

4.4. An application: Mn–Fe partitioning between garnet and orthopyroxene

Mahar et al. (1997) used element-partitioning data mainly obtained from well-equilibrated natural biotitebearing pelites, to derive thermodynamic properties for various Mn2+ end-member silicates (i.e., biotite, chlorite, cordierite, staurolite and cordierite). From a least-squares

Bhattacharya et al. (1990) studied the Mn–Fe partitioning between garnet and orthopyroxene in manganiferous granulite-facies iron formations. This exchange reaction can be formulated as:

Table 7 Values of DGR for various Mn–Mg exchange equilibria and resulting standard enthalpy of formation values, DH of , for related end-member Mn-silicates. They are based on the DGR’s and the thermodynamic properties of spessartine as given in this study (Table 5). The DGR values were derived from element partitioning data from natural mineral pairs by Mahar et al. (1997 = M97) and corrected to 4.5 kbar and 550 °C. The values are recalculated for these conditions in the column labeled HP98 using an updated version of the Holland and Powell (1998) data set. The Mn–Mg exchange equilibria are garnet-biotite (Grt-Bt), chlorite-biotite (Chl-Bt), cordierite-biotite (Crd-Bt), staurolite-biotite (St-Bt) and chloritoid-biotite (Ctd-Bt). DH of according to HP98 is shown for comparison, as well as the difference with regard to DH of of this study. So values for the various Mn end-members are also listed (from HP98). Mn–Mg exchange reaction

DGR M97 (kJ)

DGR HP98 (kJ)

Ctd-Bta St-Bt Crd-Bt Chl-Bt Grt-Bt

61.19 257.41 53.11 4.46 113.74

61.222 257.52 53.151 4.467 113.782

Mineral

DH of This study (kJ/mol)

DH of HP98 (kJ/mol)

Difference in DH of (kJ/mol)

So (J/mol  K)

Mn-Bt Mn-Ctd Mn-St Mn-Crd Mn-Chl Spessartine

5483.31 3336.10 24231.09 8695.00 7700.65 5693.16

5462.86 3329.28 24203.82 8681.36 7666.56 5646.41

20.45 6.82 27.27 13.64 34.09 46.75

433 166 1024 475 595 335

a Ctd-Bt represents the Mn–Mg exchange for: 3 Mg-Ctd + 1 Mn-Bt = 3 Mn-Ctd + 1 Mg-Bt. Other exchange equilibria are labeled analogously.

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from THERMOCALC), and assume ideal mixing, a value of ðDGoR Þ6 kbar;800 C ¼ 8674 J=mol in agreement with DGoR derived from the natural data is obtained. This gives a predicted KD = 2.644 for Mn–Fe partitioning between garnet and orthopyroxene at P = 6 kbar and T = 800 °C, compared to the natural KD = 2.646 (Fig. 11). Based on the THERMOCALC values for spessartine, ðDGoR Þ6 kbar;800 C for reaction (15) would amount to 4377 J/mol, giving a considerably different KD = 1.63. In Fig. 11 we plotted contours of constant KD for the ideal Mn–Fe exchange (15) resulting from the new spessartine properties that might be used as a geothermometer. 5. CONCLUSIONS Fig. 11. Contours of constant KD for the Mn–Fe exchange between garnet and orthopyroxene (Eq. (15)), assuming ideal mixing. Spessartine’s thermodynamic properties are from this study (Table 5), all other properties from THERMOCALC. Dot represents the P–T conditions of 6 kbar and 800 °C derived from geothermobarometry on granulite-facies rocks by Bhattacharya et al. (1990), from which manganiferous iron formations yield a natural KD = 2.65 for the Mn–Fe exchange between garnet and orthopyroxene. Dashed lines show calculated KD, if all properties, including those of spessartine, are taken from THERMOCALC.

1=3Fe3 Al2 Si3 O12 þ MnSiO3 ¼ 1=3Mn3 Al2 Si3 O12 þ FeSiO3 ;

ð15Þ

from which the relation lnðX Mn =X Fe ÞGrt ¼ lnðX Mn =X Fe ÞOpx  ðDGoR ÞP ;T =RT

ð16Þ

follows. Eq. (16) is valid, if garnet and orthopyroxene behave as ideal solid solutions, or their nonidealities mutually cancel out. A plot of ln(XMn/XFe)Opx versus ln(XMn/XFe)Grt should yield a straight line with unit slope and ðDGoR ÞP ;T =RT as intercept. From a regression of the natural partitioning data Bhattacharya et al. (1990) obtained: lnðX Mn =X Fe ÞGrt ¼ 0:941 lnðX Mn =X Fe ÞOpx þ 0:973;

The heat capacity of two different synthetic spessartine samples was measured in the temperature range 2–864 K by relaxation and differential scanning calorimetry. The standard third-law entropy of spessartine was determined for the first time from the low-temperature heat capacity measurements giving a value of So = 335.3 ± 3.8 J/mol  K. This value is considerably different from that used, for example, in the Holland and Powell (1998) thermodynamic data set. This value is based on Cp measurements made on two different spessartines synthesized in two different labs. The vibrational and magnetic contributions to heat capacity were obtained by applying the Komada and Westrum (1997) model to the measured Cp between 2 and 300 K. A magnetic transition at 6.2 K, which is probably paramagnetic/antiferromagnetic in nature, has an entropy value of 38.1 J/mol  K. This is 85% of the total maximum possible magnetic entropy given by the relation 3Rln6. The new thermodynamic properties for spessartine predict Mn–Fe partitioning between garnet and orthopyroxene in excellent agreement with KD’s resulting from natural coexisting pairs (Bhattacharya et al., 1990). ACKNOWLEDGMENTS

ð17Þ

and from geothermobarometry on the granulite terrain rocks they derived P–T conditions of P = 6 kbar and T = 800 ± 30 °C. The DGoR of exchange reaction (15) at these P–T conditions based on the natural KD data is thus 0.973*8.3145*1073 = 8682 J/mol. If we calculate ðDGoR Þ6 kbar;800 C for equilibrium (15) with spessartine properties derived in this study (Table 5, and using properties for almandine, ferrosilite and rhodonite

This work was financed by the Austrian Science Fund project number P20210-N10, National Sciences Foundation grants EAR 05-37068 and EAR 06-09967, and support of the W.C. Kelly Professorship to EJE, which are gratefully acknowledged. Prof. Marc Hirschmann is thanked for generously providing access to his high-pressure facilities at the University Minnesota and M. Grodzicki (Salzburg) is thanked for valuable discussions on magnetic properties of solids. Reviews by Giulio Ottonello, Masaki Akaogi and an anonymous reviewer helped improve the manuscript.

Calorimetric investigation of spessartine

3405

APPENDIX Raw experimental molar heat capacity, Cp, of synthetic spessartine. Numbers in parentheses are one standard deviation and refer to the last digits. M = 495.05 g/mol. The low temperature PPMS Cp’s have been measured on 29.5 (S100#20) and 24.0 mg (A627) of sample, and the DSC Cp’s on 18.5 (S100#20) and 24.4 mg (A627) of sample. The DSC data are the mean of 4–5 measurements. PPMS T (K)

DSC Cp (J/mol  K)

Sample S100#20 2.03 6.57(5) 2.04 6.58(5) 2.04 6.70(5) 2.22 8.04(6) 2.22 8.05(6) 2.22 8.12(6) 2.42 9.56(7) 2.42 9.56(7) 2.42 9.62(7) 2.64 11.18(8) 2.64 11.18(9) 2.64 11.25(8) 2.87 12.92(10) 2.87 12.92(10) 2.88 12.99(10) 3.13 14.90(11) 3.13 14.89(11) 3.14 14.97(11) 3.42 17.09(13) 3.42 17.09(13) 3.42 17.15(13) 3.72 19.50(14) 3.72 19.50(15) 3.72 19.57(15) 4.05 22.21(16) 4.05 22.23(16) 4.05 22.31(16) 4.41 25.57(18) 4.41 25.56(18) 4.41 25.67(18) 4.79 29.49(20) 4.80 29.49(20) 4.80 29.67(20) 5.22 34.60(22) 5.22 34.64(22) 5.23 34.91(21) 5.68 41.67(25) 5.68 41.75(25) 5.70 42.32(23) 6.18 53.70(34) 6.18 54.05(33) 6.18 54.06(33) 6.71 22.17(12) 6.72 22.23(11) 6.74 20.70(49) 7.28 15.42(8) 7.30 15.50(6) 7.30 15.48(6) 7.93 12.96(6) 7.94 13.00(5) 7.94 13.00(4) 8.62 11.42(5) 8.64 11.43(4) 8.64 11.42(4) 9.39 10.23(4)

T (K) 282.1 286.6 291.0 295.5 300.0 304.5 308.9 313.4 317.9 322.5 327.0 331.5 336.0 340.6 345.1 349.6 354.1 358.7 363.2 382.8 387.3 391.9 396.4 400.9 405.4 409.9 414.5 419.0 423.5 428.0 432.5 437.0 441.5 446.1 450.6 455.1 459.6 464.1 482.9 487.4 491.9 496.4 500.9 505.4 509.9 514.4 518.9 523.4 527.9 532.4 536.9 541.4 546.0 550.5 555.0

PPMS

DSC

Cp (J/mol  K)

T (K)

Cp (J/mol  K)

T (K)

325.2(26) 328.7(23) 331.8(23) 334.7(20) 338.2(27) 341.1(26) 344.2(28) 347.2(19) 349.9(15) 352.9(28) 355.8(26) 358.4(27) 361.1(26) 363.7(25) 366.4(27) 368.9(26) 371.4(27) 373.7(29) 376.1(29) 385.8(32) 387.7(29) 389.8(30) 391.9(28) 393.7(27) 395.7(28) 397.6(29) 399.5(29) 401.2(26) 402.9(23) 404.6(26) 406.5(27) 408.1(26) 409.8(27) 411.6(27) 413.3(28) 414.8(30) 416.5(32) 417.9(37) 424.1(40) 425.6(38) 426.9(36) 428.1(35) 429.3(32) 430.7(33) 431.9(34) 433.1(33) 434.3(32) 435.4(33) 436.5(34) 437.8(33) 438.8(35) 440.0(37) 441.2(37) 442.2(39) 443.2(39)

Sample A627 2.02 2.02 2.02 2.20 2.20 2.20 2.39 2.39 2.39 2.60 2.60 2.60 2.83 2.83 2.83 3.08 3.08 3.08 3.36 3.36 3.36 3.66 3.66 3.66 3.98 3.98 3.98 4.33 4.33 4.34 4.72 4.72 4.72 5.13 5.13 5.14 5.59 5.59 5.60 6.07 6.08 6.09 6.62 6.62 6.62 7.19 7.21 7.21 7.83 7.85 7.85 8.52 8.54 8.54 9.28

Cp (J/mol  K)

5.68(9) 5.72(9) 5.81(9) 7.00(10) 7.03(10) 7.11(10) 8.44(12) 8.46(12) 8.53(12) 9.94(14) 9.96(13) 10.03(13) 11.58(15) 11.60(15) 11.67(15) 13.48(17) 13.50(17) 13.58(17) 15.61(19) 15.64(19) 15.70(19) 17.97(21) 18.00(21) 18.06(21) 20.63(23) 20.65(23) 20.73(23) 23.79(25) 23.81(25) 23.93(25) 27.61(28) 27.64(28) 27.81(27) 32.41(31) 32.49(31) 32.71(30) 38.96(35) 39.07(35) 39.56(34) 47.45(50) 49.23(46) 49.76(45) 23.44(17) 23.41(16) 23.46(17) 15.25(12) 15.43(10) 15.47(10) 12.82(10) 12.96(8) 12.97(8) 11.31(8) 11.39(6) 11.38(6) 10.17(6)

282.1 326.7(15) 286.5 330.5(13) 291.0 333.9(19) 295.5 336.7(11) 299.9 339.2(9) 304.4 343.4(18) 308.9 346.5(14) 313.4 348.1(15) 317.9 352.8(12) 322.4 354.4(13) 327.0 358.3(10) 331.5 360.2(18) 336.0 363.6(9) 340.5 365.9(10) 345.1 368.3(17) 349.6 371.1(12) 354.1 373.7(11) 358.7 375.9(10) 363.2 378.4(9) 382.8 387.4(13) 387.4 389.2(19) 391.9 392.1(13) 396.4 394.0(18) 400.9 396.0(16) 405.4 397.9(14) 410.0 399.8(14) 414.5 401.7(11) 419.0 403.7(9) 423.5 405.3(7) 428.0 407.0(5) 432.5 408.7(5) 437.0 410.3(6) 441.6 412.0(4) 446.1 413.8(7) 450.6 415.4(9) 455.1 417.2(8) 459.6 418.8(8) 464.1 420.4(9) 482.9 427.0(12) 487.4 428.4(11) 491.9 429.8(11) 496.4 431.0(12) 500.9 432.1(13) 505.4 433.9(13) 509.9 435.1(10) 514.4 436.3(11) 518.9 437.4(9) 523.4 438.6(9) 527.9 439.4(6) 532.4 440.7(9) 536.9 441.7(10) 541.4 442.8(7) 545.9 443.7(8) 550.5 444.4(7) 555.0 445.7(11) (continued on next page)

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Appendix (continued) PPMS

DSC

PPMS

DSC

Cp (J/mol  K)

T (K)

Cp (J/mol  K)

Cp (J/mol  K)

T (K)

Cp (J/mol  K)

9.41 9.41 10.21 10.23 10.24 11.12 11.14 11.14 12.10 12.12 12.12 13.16 13.18 13.18 14.32 14.34 14.34 15.59 15.61 15.61 16.98 16.98 16.99 18.48 18.48 18.48 20.11 20.11 20.12 21.88 21.88 21.90 23.81 23.82 23.84 25.90 25.91 25.94 28.19 28.20 28.24 30.68 30.68 30.73 33.39 33.39 33.44 36.34 36.34 36.39 39.55 39.55 39.61 43.04 43.04 43.11 46.85

10.22(3) 10.20(3) 9.29(4) 9.27(2) 9.24(2) 8.47(3) 8.43(2) 8.42(2) 7.73(3) 7.70(1) 7.69(2) 7.08(2) 7.06(1) 7.05(1) 6.53(2) 6.49(1) 6.47(1) 6.07(2) 6.04(2) 6.02(2) 5.72(2) 5.69(2) 5.71(2) 5.50(3) 5.50(2) 5.51(3) 5.48(3) 5.49(3) 5.49(3) 5.69(4) 5.67(4) 5.69(4) 6.16(6) 6.17(6) 6.20(6) 7.01(8) 7.03(7) 7.01(8) 8.17(10) 8.17(10) 8.22(11) 9.78(13) 9.77(14) 9.82(13) 11.98(18) 11.97(18) 12.02(17) 14.72(22) 14.74(22) 14.79(22) 18.12(28) 18.13(28) 18.23(27) 22.06(32) 22.11(32) 22.19(33) 26.79(39)

559.5 564.0 582.9 587.4 591.9 596.4 600.9 605.4 609.9 614.4 618.9 623.4 627.9 632.4 637.0 641.5 646.0 650.5 655.0 659.5 664.0 683.0 687.5 692.0 696.5 701.0 705.5 710.0 714.5 719.0 723.5 728.0 732.5 737.0 741.5 746.0 750.6 755.1 759.6 764.1 783.0 787.5 792.0 796.5 801.0 805.6 810.1 814.6 819.1 823.6 828.1 832.6 837.1 841.6 846.1 850.7 855.2

444.1(37) 445.1(37) 448.5(40) 449.4(40) 450.4(41) 451.2(42) 452.2(42) 453.3(42) 454.0(40) 454.8(38) 455.9(43) 456.9(46) 457.5(43) 458.0(42) 458.6(40) 459.3(41) 460.0(41) 460.6(42) 461.4(49) 462.0(51) 462.8(50) 466.6(44) 467.1(40) 467.7(34) 468.1(29) 469.0(29) 469.5(27) 470.0(28) 470.8(25) 471.3(23) 471.9(19) 472.6(19) 473.0(23) 473.8(25) 474.3(31) 474.6(31) 474.9(38) 475.3(43) 475.7(45) 474.8(64) 480.0(59) 480.4(50) 481.9(45) 482.5(47) 483.5(51) 484.3(47) 484.9(46) 486.4(44) 486.8(50) 488.0(53) 488.9(54) 489.4(56) 489.9(61) 490.1(66) 490.5(69) 490.5(71) 490.9(68)

9.30 9.30 10.09 10.12 10.12 10.99 11.01 11.01 11.97 11.99 11.99 13.03 13.04 13.05 14.18 14.20 14.20 15.44 15.45 15.46 16.81 16.82 16.82 18.31 18.31 18.31 19.92 19.92 19.94 21.69 21.69 21.72 23.61 23.61 23.64 25.71 25.71 25.74 27.99 27.99 28.03 30.48 30.48 30.53 33.19 33.19 33.25 36.14 36.14 36.21 39.36 39.36 39.44 42.85 42.85 42.93 46.66

10.21(5) 10.21(5) 9.23(5) 9.26(4) 9.27(4) 8.43(4) 8.45(3) 8.45(3) 7.74(4) 7.75(3) 7.75(3) 7.14(3) 7.15(2) 7.14(2) 6.60(3) 6.61(2) 6.61(2) 6.18(3) 6.18(2) 6.18(2) 5.87(3) 5.86(2) 5.86(2) 5.69(3) 5.69(2) 5.68(2) 5.70(4) 5.69(3) 5.69(3) 5.91(4) 5.91(5) 5.91(4) 6.42(6) 6.42(5) 6.44(6) 7.23(7) 7.18(7) 7.36(8) 8.44(10) 8.42(9) 8.52(10) 10.04(12) 10.07(12) 10.15(13) 12.29(16) 12.29(16) 12.41(17) 15.05(20) 15.06(20) 15.21(22) 18.47(25) 18.46(26) 18.67(27) 22.46(32) 22.52(32) 22.72(34) 27.13(39)

559.5 564.0 582.9 587.4 591.9 596.4 600.9 605.4 609.9 614.4 618.9 623.4 627.9 632.4 637.0 641.5 646.0 650.5 655.0 659.5 664.0 683.0 687.5 692.0 696.5 701.0 705.5 710.0 714.5 719.0 723.5 728.0 732.5 737.0 741.6 746.1 750.6 755.1 759.6 764.1 783.0 787.5 792.0 796.5 801.1 805.6 810.1 814.6 819.1 823.6 828.1 832.6 837.1 841.6 846.2 850.7 855.2

446.7(9) 448.0(11) 451.2(7) 452.0(9) 452.7(10) 453.8(13) 454.6(13) 455.3(15) 455.7(14) 457.0(13) 457.7(8) 458.2(16) 459.2(11) 459.8(12) 460.4(8) 461.0(9) 462.0(9) 462.3(13) 463.4(20) 464.0(11) 465.4(12) 468.1(14) 468.8(10) 469.4(9) 469.9(12) 470.7(10) 471.0(8) 470.7(11) 472.2(12) 472.7(21) 473.6(18) 474.3(20) 474.8(16) 475.2(15) 475.9(19) 476.8(16) 477.2(18) 477.7(16) 477.7(17) 479.0(13) 481.5(26) 482.3(35) 483.1(33) 483.3(45) 485.0(51) 485.6(56) 486.3(64) 487.2(63) 487.7(72) 487.5(48) 489.7(75) 490.1(78) 490.5(70) 491.1(69) 492.2(70) 491.5(74) 490.2(21)

46.85 46.92 50.99 50.99

26.82(39) 26.92(40) 32.24(46) 32.22(46)

859.7 864.2

490.6(71) 490.6(75)

46.66 46.75 50.81 50.81

27.14(39) 27.43(42) 32.79(47) 32.81(47)

859.7 864.2

492.9(56) 492.7(59)

T (K)

T (K)

Calorimetric investigation of spessartine

3407

Appendix (continued) PPMS

DSC

PPMS

T (K)

Cp (J/mol  K)

T (K)

Cp (J/mol  K)

51.07 55.50 55.50 55.58 60.42 60.42 60.51 65.78 65.78 65.88 71.61 71.61 71.71 77.94 77.95 78.04 84.84 84.85 84.94 92.35 92.35 92.45 100.50 100.50 100.60

32.44(48) 38.50(57) 38.52(55) 38.75(57) 45.72(64) 45.66(64) 46.00(67) 53.87(74) 54.02(74) 54.28(77) 62.90(82) 63.08(83) 63.42(88) 73.40(95) 72.75(94) 72.93(97) 84.52(105) 84.98(105) 84.98(109) 96.39(115) 96.36(115) 96.52(118) 109.39(129) 110.00(128) 109.74(132)

50.90 55.33 55.33 55.43 60.25 60.25 60.35 65.60 65.60 65.70 71.43 71.43 71.53 77.76 77.76 77.88 84.67 84.67 84.80 92.18 92.18 92.30 100.36 100.37 100.50

33.08(50) 39.26(55) 39.26(55) 39.62(60) 46.61(65) 46.62(65) 46.87(68) 54.90(75) 54.86(75) 55.23(78) 64.39(84) 64.43(85) 65.40(98) 74.13(96) 73.93(95) 74.13(99) 84.34(106) 84.52(105) 84.98(110) 96.44(116) 96.23(118) 96.67(120) 109.95(126) 110.01(127) 110.42(132)

109.41 109.41 109.51 119.10 119.10 119.21 129.66 129.66 129.78 141.16 141.16 141.28 153.66 153.66 153.77 167.27 167.28 167.37 182.10 182.10 182.18 198.30 198.31 198.37 215.84 215.84 215.88 234.91 234.94 234.94 255.65 255.73 255.74 278.17

123.31(139) 123.61(142) 124.08(144) 139.18(148) 139.19(151) 139.67(154) 155.70(161) 155.82(160) 156.13(165) 173.13(170) 173.13(170) 174.85(172) 190.73(176) 190.80(177) 191.31(182) 209.41(186) 208.91(185) 209.47(192) 227.25(192) 227.66(192) 227.98(201) 249.48(199) 249.85(197) 249.72(205) 269.32(204) 269.10(205) 269.28(214) 286.83(220) 286.95(211) 286.93(210) 306.17(221) 305.31(215) 305.67(214) 325.10(227)

109.27 109.27 109.40 118.96 118.96 119.10 129.53 129.53 129.67 141.02 141.02 141.16 153.53 153.54 153.66 167.16 167.16 167.30 182.11 182.12 182.25 198.25 198.25 198.37 215.80 215.81 215.88 234.90 234.91 234.94 255.68 255.70 255.71 278.22

125.29(139) 125.35(139) 125.39(143) 141.28(148) 141.74(149) 142.04(153) 157.44(157) 157.74(158) 158.20(163) 173.95(167) 174.33(166) 174.77(173) 192.62(174) 192.60(175) 192.43(175) 212.82(182) 212.60(182) 213.27(190) 231.58(190) 230.95(188) 232.02(198) 251.44(196) 251.43(196) 251.69(204) 271.47(204) 271.18(203) 271.71(215) 288.89(209) 288.76(209) 288.89(217) 308.06(220) 308.62(213) 308.48(213) 328.27(225)

T (K)

Cp (J/mol  K)

DSC T (K)

Cp (J/mol  K)

(continued on next page)

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Appendix (continued) PPMS

DSC

T (K)

Cp (J/mol  K)

278.33 278.33 302.91 302.93 303.07

324.52(223) 324.54(224) 342.69(227) 343.16(226) 343.04(230)

T (K)

PPMS Cp (J/mol  K)

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DSC

T (K)

Cp (J/mol  K)

278.32 278.33 302.74 302.93 302.95

328.84(222) 327.95(221) 345.11(227) 345.27(226) 345.70(226)

T (K)

Cp (J/mol  K)

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