Experimentally Determined Standard Thermodynamic Properties of Synthetic MgSO 4 ·4H 2 O (Starkeyite) and MgSO 4 ·3H 2 O: A Revised Internally Consistent Thermodynamic Data Set for Magnesium Sulfate Hydrates

Abstract

The enthalpies of formation of synthetic MgSO₄·4H₂O (starkeyite) and MgSO₄·3H₂O were obtained by solution calorimetry at T=298.15 K. The resulting enthalpies of formation from the elements are \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\Delta_{ \rm f}H^0_{298}$$\end{document} (starkeyite)=−2498.7±1.1 kJ·mol⁻¹ and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\Delta_{ \rm f}H^0_{298}$$\end{document} (MgSO₄·3H₂O)=−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: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$S^0_{298}$$\end{document} (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 MgSO₄·nH₂O 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. Key Words: Mg sulfates—Starkeyite—Thermodynamic data—Entropy—Enthalpy—Calorimetry. Astrobiology 12, 1042–1054.

1. Introduction

A number of different hydrated forms of MgSO₄·nH₂O (1≤n≤11) as well as three different polymorphs of the anhydrous MgSO₄ have been described from laboratory experiments (Table 1). Most of them have been found as minerals on Earth (Peterson, 2011), and recent space missions support the presence of significant quantities of Mg sulfates in soils and sediments of Mars (cf. e.g., Chipera and Vaniman, 2007; Roach et al., 2009 and references therein). Recently, the enthalpies of formation from the elements ( \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\Delta_{ \rm f}H^0_{298}$$\end{document} ) of kieserite (n=1), sanderite (n=2), hexahydrite (n=6), and epsomite (n=7) were measured (Grevel and Majzlan, 2009). Subsequently, these authors combined their enthalpy data with other experimental studies on the stability of several Mg-sulfate hydrates, which included epsomite, hexahydrite, starkeyite (n=4), and kieserite, as a function of temperature and relative humidity (van't Hoff et al., 1912; Chou and Seal, 2003, 2007)1 in a rigorous mathematical programming (MAP) analysis to obtain an internally consistent thermodynamic database for the system MgSO₄–H₂O. This resulted in the phase diagram shown in Fig. 1. As concluded by Grevel and Majzlan (2009), starkeyite is metastable with respect to kieserite formation, but kieserite formation is known to be very sluggish under typical dehydration conditions (Steiger et al., 2011). Therefore, Fig. 1 shows an extended field of metastable existence of starkeyite, which makes this phase a major reaction product in dehydration reactions. However, the data for starkeyite in the database of Grevel and Majzlan (2009) were derived from estimates; therefore, direct measurements for this phase were strongly desired which defines the major aim of the present study. We have obtained \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\Delta_{ \rm f}H^0_{298}$$\end{document} of synthetic starkeyite by solution calorimetry in water at room temperature. The standard entropy of starkeyite was derived from low-temperature heat capacity measurements acquired with a physical property measurement system (PPMS). Additionally, the enthalpy of formation of the trihydrate (MgSO₄·3H₂O) was measured.

FIG. 1.

Calculated stability of Mg-sulfate hydrates as a function of temperature and relative humidity (Grevel and Majzlan, 2009). The size of the large symbols representing the experimentally obtained reversals for the reactions epsomite=hexahydrite + H₂O (Chou and Seal, 2003), hexahydrite=starkeyite + 2H₂O (Chou and Seal, 2007), and hexahydrite=kieserite + 5H₂O (van't Hoff et al., 1912) indicates the experimental uncertainty. Starkeyite is metastable. The solubility curve was calculated after Archer and Rard (1998) based on solubility data of Linke (1965). Small symbols represent new measurements of equilibrium relative humidities for several hydration–dehydration equilibria in the MgSO₄–H₂O system (Steiger et al., 2011). Only data points which were used by Steiger et al. (2011) to derive their final phase diagram are shown.

Table 1.

Different Hydrated Forms of MgSO₄·nH₂O (0≤n≤11) as Described in the Literature

Phase	n	Reference ^a
α-MgSO₄	0	Rentzeperis and Soldatos, 1958; Fortes et al., 2007
β-MgSO₄	0	Coing-Boyat, 1962; Yamaguchi and Kato, 1972; Fortes et al., 2007
γ-MgSO₄	0	Rowe et al., 1967; Daimon and Kato, 1984
Kieserite	1	Hawthorne et al., 1987
Reagent monohydrate^b	1	Chipera and Vaniman, 2007; Grindrod et al., 2010; Steiger et al., 2011
5/4 hydrate^b	1.25	van't Hoff and Dawson, 1899; Hodenberg and Kühn, 1967
Sanderite	2	Ma et al., 2009a
2.4 hydrate^c	2.4	Chipera and Vaniman, 2007
2.5 hydrate^c	2.5	Ma et al., 2009b
Trihydrate	3	Hodenberg and Kühn, 1967; Fortes et al., 2010
Starkeyite	4	Baur, 1962, 1964a
Cranswickite	4	Peterson, 2011
Pentahydrite	5	Baur and Rolin, 1972
Hexahydrite	6	Zalkin et al., 1964
Epsomite	7	Baur, 1964b; Ferraris et al., 1973; Fortes et al., 2006
Meridianiite	11	Peterson and Wang, 2006; Peterson et al., 2007; Fortes et al., 2008

In case the structure is known, this reference is given.

The reagent monohydrate described by Chipera and Vaniman (2007) is reported to be a mixture of the 5/4 hydrate and sanderite (Grindrod et al., 2010).

The 2.5 hydrate described by Ma et al. (2009b) is the 2.4 hydrate reported by Chipera and Vaniman (2007) (cf. Ma et al., 2009b).

Steiger et al. (2011) reported new measurements of equilibrium relative humidities for several stable and metastable hydration–dehydration equilibria in the MgSO₄–H₂O system (see Fig. 1). These measurements served as additional constraints for our MAP procedure to obtain a revised internally consistent thermodynamic data set for magnesium sulfate hydrates.

2. Materials

2.1. Starkeyite

Starkeyite was produced at the Department of Chemistry (University of Hamburg, Germany) by dehydration of MgSO₄·6H₂O, (Fluka 00627 p.a.) in a drying cabinet (4 days) at about 320 K (Steiger et al., 2011). After verification by X-ray powder diffraction (XRD), the powdered material was kept in a closed vial in the drying cabinet before it was sent to Bochum by regular mail. XRD patterns were obtained on a Stadi P powder diffractometer (STOE, Germany) with transmission configuration for which Cu-Kα radiation was used. The XRD patterns were compared to JCPDS standard 24-720. Additionally, powder XRD measurements acquired with a Siemens D 500 diffractometer with 0.02° step scans every 2 s between 3° and 65° 2θ were collected before and after the C_p measurements in Salzburg.

2.2. The trihydrate

Crystals of MgSO₄·3H₂O were prepared by gentle evaporation of a 25 wt % MgSO₄ solution at 105°C for ∼48–72 h without stirring (Fortes and Lemée-Cailleau, 2010). The crystals grown in this manner are clusters of euhedral colorless plates, 1–2 mm in length and 0.1–0.5 mm in thickness (Fig. 2). After extraction from the solution, the crystals of MgSO₄·3H₂O became somewhat dull. However, batches of crystals, far from being unstable as reported by Hodenberg and Kühn (1967), were preserved in a small glass bottle for a period of ∼10 days and withstood repeated exposure to warm air (∼30°C) and examination under a microscope. Thus, some crystal fragments could be transported to Jena by mail for use in the calorimetry experiments. Once powdered under air, this phase hydrates extremely rapidly, so measurements requiring powder samples were not carried out on this material.

FIG. 2.

Synthetic MgSO₄·3H₂O typically forms small clusters of blades (each shown here is ∼1 mm across the 010 face).

The crystal structure of the deuterated analogue MgSO₄·3D₂O was previously determined from neutron single-crystal diffraction measurements at the Institut Laue-Langevin, Grenoble, France (Fortes and Lemée-Cailleau, 2010), and the thermal expansion behavior was measured by high-resolution neutron powder diffraction at the ISIS neutron spallation source (Fortes et al., 2010). The room-temperature unit-cell dimensions, a=8.1925(2) Å, b=10.9210(2) Å, c=12.3866(4) Å and space-group Pbca, agree with the values previously published by Hodenberg and Kühn (1967) for the protonated form (MgSO₄·3H₂O) that was used in this work.

3. Calorimetric Methods

3.1. Solution calorimetry

For solution calorimetry, we used the commercial IMC-4400 isothermal microcalorimeter (Calorimetry Sciences Corporation) described by Ackermann et al. (2009) located at the Institute of Geosciences (Friedrich-Schiller University, Jena). The calorimetric solvent (25 mL of deionized water) contained in a closed polyether ether ketone cup with a total volume of 60 mL was inserted into the liquid bath of the calorimeter, which was kept at a constant temperature of 25°C (±0.0005°C). During the stabilization (∼8 h) and the experiment (80–100 minutes), the solvent was stirred with a SiO₂ glass stirrer driven by a motor positioned about 40 cm from the active zone of the instrument. The fine-grained starkeyite sample was pressed into pellets of about 20 mg and weighed on a microbalance with a precision of 0.002 mg (as stated by the manufacturer). In contrast, crystal fragments of the coarse-grained trihydrate sample were weighed out and used for calorimetry experiments without grinding. Directly after the weighing, the samples were dropped through a SiO₂ glass tube into the solvent, and the heat produced or consumed during the dissolution was measured. The solution was always freshly prepared, and the final molalities of Mg²⁺ and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${ \rm SO}_4^{2 - }$$\end{document} were identical in each run. No dilution (or concentration) effects need to be considered in this type of calorimetry.

The heat flow between the reaction cup and the constant temperature reservoir was then integrated to calculate the caloric effect. The calorimeter was calibrated by dissolving ∼20 mg pellets of KCl in 25 g of deionized water. Prior to each calibration measurement, the potassium chloride was heated overnight in the furnace at 500°C to remove any adsorbed water. The expected heat effect for the calibration runs was calculated from Parker (1965).

3.2. Thermogravimetry

The weight loss of two starkeyite samples (∼50 mg each) was determined with a STA 503 Thermal Analyzer (BÄHR-Thermoanalyse GmbH) located at Ruhr University Bochum. The samples were investigated under N₂-enriched air (1 atm) in the temperature range from room temperature to 600°C at a heating rate of 5°C/min. Differential thermal analysis (DTA) and thermogravimetric analysis (TG) signals were recorded simultaneously.

3.3. Low-temperature heat capacity measurements

The low-temperature (5–300 K) heat capacity behavior of starkeyite was investigated with the Physical Property Measurement System (PPMS) constructed by Quantum Design (Lashley et al., 2003) installed at Salzburg University. About 20 mg of the powdered sample was sealed in an Al pan so that the powder was compressed into an ∼1 mm thick layer (Dachs and Bertoldi, 2005). Heat capacity was measured at 60 different temperatures and three times at each temperature on cooling from 300 K with a logarithmic spacing. A full PPMS experiment to measure C_p consisted of an “addenda run” and a “sample run.” During the first measurement, the heat capacity of the empty sample platform was determined. In the second measurement, the sample was placed in the pan, and the heat capacity of the whole ensemble was measured. The net heat capacity of the sample is then given by the difference between the two measurements. An uncertainty of ±0.02 mg for the sample weight was adopted for converting the PPMS data from units of μJ·K⁻¹ to units of J·K⁻¹·mol⁻¹.

3.4. Differential scanning calorimetry

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 (Dachs and Benisek, 2011). The powdered sample was put into Perkin Elmer Al pans covered with lids. The measurements were performed under a flow of Ar gas, and the calorimeter block was kept at a constant temperature of 243.3 K with a Perkin Elmer Intracooler. The growth of ice crystals on the calorimeter block was prevented by a flow of dried air (200 mL·min⁻¹). Additionally, the cover heater was turned off.

The complete investigation consisted of three separate measurements: a blank, a reference, and a sample measurement. Before the sample measurement, the DSC was calibrated with a reference run by using a synthetic single crystal of corundum (31.764 mg) whose heat capacity values were taken from the National Bureau of Standards Certificate (Ditmars et al., 1982). Two measurement series were performed.

4. Experimental Results

4.1. Enthalpy measurements

Five experiments were carried out for starkeyite, four for the trihydrate crystals. After the first two experiments with the trihydrate, the crystal pieces became whitish, and the measured solution enthalpies deviated progressively from the first two results. Apparently, a significant volume of the crystals started hydrating; therefore, the last two results were discarded. The thermodynamic cycle listed in Table 2 was used to calculate the enthalpy of formation from the obtained enthalpies of dissolution of both studied hydrates of MgSO₄.

Table 2.

Thermodynamic Cycle to Calculate \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\Delta_{ \rm f}{\rm H}^0_{298}$$ \end{document} (MgSO₄·nH₂O), n=3, 4

Reaction		Enthalpy (kJ·mol⁻¹)
1	α-MgSO₄ (cr)= \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$[ { \rm Mg}^{2 + } + { \rm SO}_4^{2 - } ]$$\end{document} (aq)	−85.52^a±0.91^b (3)^c
2_starkeyite	MgSO₄·4H₂O (cr)= \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$[ { \rm Mg}^{2 + } + { \rm SO}_4^{2 - } + 4{ \rm H}_2 { \rm O} ]$$\end{document} (aq)	−18.45±0.17 (5)
2_trihydrate	MgSO₄·3H₂O (cr)= \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$[ { \rm Mg}^{2 + } + { \rm SO}_4^{2 - } + 3{ \rm H}_2 { \rm O} ]$$\end{document} (aq)	−21.01±0.79 (2)
3	H₂O (l)=H₂O (aq)	0
4	Mg (cr) + S (cr) + 2O₂ (g)=α-MgSO₄ (cr)	−1288.5±0.5^d
5	H₂ (g) + ½O₂ (g)=H₂O (l)	−285.8±0.1^e
6_starkeyite	Mg (cr) + S (cr) + 4O₂ (g) + 4H₂ (g)=MgSO₄·4H₂O (cr)	−2498.7±1.1
6_trihydrate	Mg (cr) + S (cr) + 3.5O₂ (g) + 3H₂ (g)=MgSO₄·3H₂O (cr)	−2210.3±1.3

Average.

Two standard deviations of the mean.

Number of measurements.

Ko and Daut (1979), DeKock (1986).

Robie and Hemingway (1995).

Our measured \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\Delta_{ \rm f}H^0_{298}$$\end{document} value for starkeyite, −2498.7±1.1 kJ·mol⁻¹, is about 2 kJ·mol⁻¹ more negative than the estimates reported in the literature (Wagman et al., 1982; DeKock, 1986; Pabalan and Pitzer, 1987; cf. Grevel and Majzlan, 2009). For the trihydrate, the derived enthalpy of formation from the elements equals −2210.3±1.3 kJ·mol⁻¹. Assuming a linear correlation between the number of water molecules and the enthalpy of formation of the respective Mg-sulfate hydrates as can be seen in Fig. 3, the enthalpy of formation from the elements of a Mg-sulfate hydrate with n water molecules can be roughly estimated as \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} \Delta_{ \rm f}H^0_{298} ( { \rm MgSO}_4 \cdot n{ \rm H}_2{ \rm O}) = ( - 298.33n - 1302.70 ) \, { \rm kJ \cdot mol}^{ - 1} \tag{1}\end{align*}\end{document}

FIG. 3.

Linear correlation between the number of water molecules and the enthalpy of formation of Mg-sulfate hydrates. The measured values given in the figure for kieserite (n=1), sanderite (n=2), hexahydrite (n=6), and epsomite (n=7) were taken from Grevel and Majzlan (2009); for starkeyite (n=4) and the trihydrate (n=3) see this study. The value for anhydrous Mg sulfate (n=0) was taken from Ko and Daut (1979).

with a high correlation coefficient of R ²=0.99. This high value tempts to consider the estimated formation enthalpies for the Mg-sulfate hydrates to be very good. It has to be noted that the correlation is so high because formation enthalpies from elements (not formation enthalpies from oxides or formation enthalpies from MgSO₄ and H₂O) are considered. The estimated enthalpies of formation of the respective magnesium sulfate hydrates differ from the experimentally determined properties by 2.7 kJ·mol⁻¹ for starkeyite to more than 14 kJ·mol⁻¹ for the anhydrous magnesium sulfate. For the delicate phase equilibria considered in this paper, such difference would mean a shift of relative humidity of an equilibrium reaction at a given temperature of more than 100%, thus shifting some curves completely out of the T-RH space. So, estimates such as these may be used as a first guess but will most likely not lead to correct phase diagrams.

Using Eq. 1 for obtaining an estimate for \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\Delta_{ \rm f}H^0_{298}$$\end{document} (meridianiite) (n=11) results in −4584.3 kJ·mol⁻¹, as compared to the estimate given by Chou and Seal (2007), −4579.8 kJ·mol⁻¹. The difference between the two values, ∼4.5 kJ·mol⁻¹, documents well the pitfalls of simple linear fits to thermodynamic data.

4.2. Thermogravimetry

Two different batches of the powdered starkeyite sample were analyzed. The first batch was untreated. The other batch was finely ground, used for the C_p measurements in Salzburg, and analyzed afterwards. The untreated batch weighed 57.50 mg, the ground sample 46.75 mg. Figure 4 shows the results for the ground sample. The results of the untreated sample are almost equivalent. Starkeyite loses water in five steps; the total loss was calculated to be 36.9% for the ground sample and 37.0% for the untreated material. These observations agree well with the theoretical value of 37.4%. Thus, we conclude that grinding had no effect on the state of hydration of our sample. Analyzing the TG curve (Fig. 4), we concluded that starkeyite dehydrates to sanderite (n=2) in two steps. The observed loss of water, 18.6%, agrees very well with the theoretical value, 18.7%. In the next step, dehydration to kieserite occurs—27.8% weight loss was observed versus 28.1% weight loss theoretically. The final dehydration of kieserite seems to occur in two more steps.

FIG. 4.

TG/DTA analysis of starkeyite. The sample weight was 46.75 mg. The total mass loss equals 36.9%. Obviously, starkeyite loses water in five steps; the weight loss of each single step is shown in the TG diagram.

4.3. Entropy and heat capacity of starkeyite

A single PPMS measurement on 21.618 mg of the ground starkeyite sample was carried out. The sample, which was verified to be starkeyite by XRD directly before and after the measurement, had lost 0.124 mg of its weight during the experiment, which was taken into consideration in the data analysis. All measured heat capacities for starkeyite are listed in Table 3 and shown in Fig. 5; the low-temperature PPMS data can be found in Table 3a, the DSC data (around ambient T) in Table 3b. Smoothed thermodynamic functions for starkeyite (Table 3c) were calculated by fitting this final C_p data set to a combination of Debye, Einstein, and Schottky functions (e.g., Boerio-Goates et al., 2002; Dachs et al., 2010) and extrapolating this function to 0 K: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*}C_p = 3R ( n_d \cdot D ( \theta_D ) + n_e \cdot E ( \theta_E ) + n_s \cdot S ( \theta_S ) ) \tag{2}\end{align*}\end{document}

Table 3a.

Raw Experimental Low-Temperature Molar Heat Capacities, C _p, of Starkeyite Obtained from the PPMS Experiments

T (K)	C_p (J·K⁻¹·mol⁻¹)	σ-C_p (J·K⁻¹·mol⁻¹)
5.067	0.059	0.00215
5.434	0.073	0.00182
5.828	0.091	0.00105
6.248	0.113	0.00227
6.705	0.142	0.00215
7.159	0.174	0.00163
7.671	0.217	0.00201
8.223	0.272	0.00242
8.813	0.342	0.00290
9.443	0.429	0.00353
10.118	0.528	0.03575
10.853	0.683	0.00439
11.629	0.861	0.00537
12.460	1.085	0.00652
13.352	1.364	0.00790
14.308	1.703	0.00974
15.332	2.130	0.01221
16.433	2.653	0.01441
17.610	3.277	0.01805
18.874	4.031	0.02247
20.220	4.937	0.02663
21.671	6.015	0.06166
23.225	7.256	0.05951
24.889	8.720	0.04753
26.680	10.420	0.06518
28.598	12.401	0.06576
30.655	14.617	0.07856
32.859	17.168	0.09122
35.223	20.024	0.11178
37.758	23.191	0.13084
40.474	26.740	0.15821
43.385	30.568	0.21071
46.502	34.702	0.20165
49.843	39.147	0.22764
53.427	43.985	0.25872
57.273	49.196	0.54695
61.403	54.461	0.32973
65.825	60.198	0.46813
70.558	66.212	0.41880
75.627	72.553	0.43399
81.056	79.155	0.49775
86.875	85.765	0.53520
93.112	92.661	0.57135
99.812	100.08	0.60159
106.99	107.78	0.91490
114.68	115.25	0.88310
122.94	123.93	0.73427
131.78	132.17	0.90458
141.28	141.61	1.2296
151.44	151.08	0.88392
162.40	161.38	1.2763
174.06	171.53	1.1671
186.55	181.95	1.0242
199.94	193.06	1.0749
214.26	203.97	1.6468
229.63	214.97	1.7883
246.10	226.99	1.0373
263.71	238.54	1.0712
282.55	250.61	1.1671
302.62	263.83	1.1103

Table 3b.

Raw Experimental DSC Molar Heat Capacities, C _p, of Starkeyite around Ambient Temperature

T (K)	C_p (J·K⁻¹·mol⁻¹)	σ-C_p (J·K⁻¹·mol⁻¹)
281.91	250.96	0.5823
286.39	253.68	0.4351
290.86	256.48	0.4166
295.35	259.37	0.4786

The data are the mean of two series of measurements.

Table 3c.

Smoothed Molar Thermodynamic Functions of Starkeyite at Temperatures between 5 and 298.15 K (M=192.4296 g mol ^{− 1}) Calculated from Eq. 2; Debye, Einstein, and Schottky Parameters Are Listed in Table 3d

T (K)	C_p (J·K⁻¹·mol⁻¹)	S (J·K⁻¹·mol⁻¹)	(H(T) − H(0))/T (J·K⁻¹·mol⁻¹)	−(G(T) − H(0))/T (J·K⁻¹·mol⁻¹)
5	0.052635	0.016892	0.012723	0.004169
6	0.096049	0.030039	0.02273	0.0073089
7	0.16035	0.049382	0.037526	0.011856
8	0.24934	0.076306	0.058163	0.018143
9	0.36747	0.11218	0.085674	0.026509
10	0.52024	0.15848	0.12118	0.037298
11	0.71363	0.21679	0.16592	0.050867
12	0.95314	0.2888	0.2212	0.067593
13	1.2431	0.37619	0.28832	0.087866
14	1.5864	0.48055	0.36846	0.11208
15	1.9845	0.60326	0.46262	0.14063
20	4.7853	1.5261	1.1657	0.36047
25	8.8223	3.0063	2.2742	0.73214
30	13.899	5.0496	3.7758	1.2738
35	19.750	7.6236	5.6326	1.9910
40	26.404	10.691	7.8089	2.8817
45	33.080	14.187	10.247	3.9401
50	39.646	18.013	12.859	5.1537
60	52.529	26.384	18.401	7.9831
70	65.173	35.434	24.183	11.252
80	77.486	44.945	30.080	14.865
90	89.300	54.760	36.009	18.750
100	100.52	64.755	41.905	22.850
110	111.16	74.840	47.721	27.119
120	121.29	84.949	53.433	31.515
130	131.01	95.043	59.029	36.014
140	140.42	105.10	64.509	40.590
150	149.59	115.10	69.876	45.224
160	158.56	125.04	75.139	49.902
170	167.35	134.92	80.306	54.613
180	175.96	144.73	85.381	59.348
190	184.40	154.47	90.372	64.098
200	192.63	164.14	95.280	68.859
210	200.65	173.73	100.11	73.625
220	208.43	183.25	104.86	78.393
230	215.96	192.68	109.52	83.155
240	223.23	202.02	114.11	87.914
250	230.23	211.28	118.62	92.664
260	236.96	220.44	123.04	97.402
270	243.40	229.51	127.38	102.13
280	249.57	238.47	131.63	106.84
290	255.46	247.33	135.80	111.53
298.15	260.07	254.48	139.14	115.34

Table 3d.

Debye, Einstein, and Schottky Parameters (Eq. 2) Obtained from the Fit to the Heat Capacity Data; T ^switch=35.68 K

T-range	n_d	θ_D	n_e	θ_E	n_s	θ_S
T<T ^switch	4.0800	271.3269	0.3346	100.1249	0.0180	47.5581
T>T ^switch	7.9937	460.8215	7.7371	1013.3241	2.0108	146.9463

where D(θ_D), E(θ_E), and S(θ_S) are the Debye, Einstein, and Schottky functions, respectively, and n_d, n_e, n_s, θ_D, θ_E, and θ_S are adjustable parameters. For an optimal representation of the experimental low-temperature C_p behavior, the data were divided into low- and high-temperature intervals. Thus, two sets of fit parameters are listed in Table 3d, which have to be used below and above T ^switch, respectively. As can be seen from Fig. 5, the agreement between PPMS and DSC data around ambient temperature is very good; at 295 K, the latter are larger by 0.13%. As expected for the simple compound, no anomalies in the C_p data are observed.

FIG. 5.

Heat capacity of starkeyite. The solid line represents a Maier-Kelley fit to all C_p data between 229 and 303 K.

The resulting standard entropy of starkeyite, 254.48±2.0 J·K⁻¹·mol⁻¹, is very close to the estimate of Pabalan and Pitzer (1987), 254.76 J·K⁻¹·mol⁻¹, and also close to the optimized value 259.9 J·K⁻¹·mol⁻¹ from Grevel and Majzlan (2009). Obviously, the earlier estimate of 246.86 J·K⁻¹·mol⁻¹ (DeKock, 1986) can be discarded.

5. Refinement of the Thermodynamic Data

All Mg-sulfate hydrates change their hydration state in response to the local temperature and humidity conditions. In their case study, Grevel and Majzlan (2009) performed a MAP analysis based on a simultaneous treatment of calorimetric data and constraints given from relevant reversal experiments (van't Hoff et al., 1912; Chou and Seal, 2003, 2007) according to the hydration-dehydration reaction \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*}{ \rm MgSO}_4 \cdot n { \rm H}_2{ \rm O} \leftrightarrow { \rm MgSO}_4 \cdot m{ \rm H}_2{ \rm O} + ( n - m ) { \rm H}_2{ \rm O} , n > m \tag{3}\end{align*}\end{document}

For this reaction, the standard Gibbs free energy (in kJ·mol⁻¹) is given as \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} \Delta G_{\rm r}^0 = \Delta H_{\rm r}^0 - T \Delta S_{\rm r}^0 & = - {\rm R}T \ {\rm ln} \ K \\& = - (n - m) \{\rm R}T \ {\rm ln} \ a_{{\rm H}_{2}}{O} \\& = - (n - m) \ {\rm R}T \ {\rm ln} \ f^*_{{\rm H}_{2}}{O} - ( n - m ) \ \\& \quad{\rm R}T \ {\rm ln} \ (\% {\rm RH}/100) \quad\quad\quad(4) \end{align*}\end{document}

where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\Delta H_{ \rm r}^0$$\end{document} , standard enthalpy of reaction; \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\Delta S_{ \rm r}^0$$\end{document} , standard entropy of reaction; K, equilibrium constant; R=8.31451 J·mol⁻¹·K⁻¹, gas constant; T, temperature in K; \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$a_{{ \rm H}_{2}{ \rm O}}$$\end{document} , activity of H₂O in vapor; \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$f^*_{{ \rm H}_{2}{ \rm O}} = f_{{ \rm H}_{2}{ \rm O}} / f^0_{{ \rm H}_{2}{ \rm O}} , f_{{ \rm H}_{2}{ \rm O}}$$\end{document} fugacity of pure H₂O in bar, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$f^0_{{ \rm H}_{2}{ \rm O}}$$\end{document} bar; %RH, relative humidity.

Mathematical programming deals with a system of inequalities given by

(1) phase equilibria constraints resulting from the reversal experiments, and

(2) phase property constraints gained from calorimetry.

According to Reaction 3, the low-temperature half-bracket indicates the T-%RH condition at which the higher-hydrate sulfate, that is, the reactant, has proven to be stable—here \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\Delta G^0_{ \rm r} + { \rm R}T \ { \rm ln} \ K > 0$$\end{document} . On the other hand, the other half-bracket denotes the relative stability of the lower-hydrate sulfate plus water—that means \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\Delta G^0_{ \rm r} + { \rm R}T \ { \rm ln} \ K < 0$$\end{document} . In terms of Eq. 4: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} \Delta G^0_{\rm r} & + {\rm R}T \ {\rm ln} \ K = \Delta H_{\rm r}^0 - T \Delta S_{\rm r}^0 + ( n - m ) \\& [{\rm R}T \ {\rm ln} \ f^*_{\rm H_{2}}{O} + {\rm R}T \ {\rm ln} \ (\% {\rm RH} / 100)] > 0 \\& ({\rm reactant} \ {\rm stable}) (5{\rm a})\end{align*}\end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} \Delta G^0_{ \rm r} & + { \rm R}T \ { \rm ln} \ K = \Delta H_{ \rm r}^0 - T \Delta S_{ \rm r}^0 + ( n - m ) \\& [ { \rm R}T \ { \rm ln} \ f^*_{\rm H_{2}O} + { \rm R}T \ { \rm ln} \ ( \% { \rm RH} / 100 ) ] < 0 \\& ({\rm products} \ {\rm stable})(5{\rm b}) \end{align*}\end{document}

These inequalities can be rewritten as \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} & \Delta_{ \rm f}H_{298}^0 ( { \rm MgSO}_4 \cdot m{ \rm H}_2{ \rm O} ) - \Delta_{ \rm f}H_{298}^0 ( { \rm MgSO}_4 \cdot n{ \rm H}_2{ \rm O} ) \\& - T ( S_{298}^0 ( { \rm MgSO}_4 \cdot m{ \rm H}_2{ \rm O} ) - S_{298}^0 ( { \rm MgSO}_4 \cdot n{ \rm H}_2{ \rm O} ) ) + { \rm r} > 0 ( 6{ \rm a} ) \end{align*}\end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} & \Delta_{ \rm f}H_{298}^0 ( { \rm MgSO}_4 \cdot m{ \rm H}_2{ \rm O} ) - \Delta_{ \rm f}H_{298}^0 ( { \rm MgSO}_4 \cdot n{ \rm H}_2{ \rm O} ) \\& - T ( S_{298}^0 ( { \rm MgSO}_4 \cdot m{ \rm H}_2{ \rm O} ) - S_{298}^0 ( { \rm MgSO}_4 \cdot n{ \rm H}_2{ \rm O} ) ) + { \rm r} < 0 ( 6{ \rm b} ) \end{align*}\end{document}

r denoting the sum of the remaining terms of Eq. 5, which are kept constant throughout the MAP procedure: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} \begin{split}{\rm r} = &{_{_{_{_{298}}}}\int^T} \Delta C_p {\rm d}T + T\cdot{_{_{_{_{298}}}}\int^T} \Delta C_p / T \ { \rm d}T \\& + ( n - m ) \{ [ \Delta_{ \rm f} H_{298}^0 ( { \rm H}_2{ \rm O} ) - TS_{298}^0( { \rm H}_2{ \rm O} ) ] \\& + { \rm R}T \ { \rm ln} \ f^*_{{ \rm H}_{2}{ \rm O}} + { \rm R}T \ { \rm ln} \ ( \% {\rm RH} / 100)\} \end{split} \tag{7}\end{align*}\end{document}

Grevel and Majzlan (2009) assumed ΔC_p =0 in their MAP calculations because only a limited amount of C_p data of the different hydrates of MgSO₄ was available. The magnitude of the error introduced by this assumption is not known exactly but is likely not large, because the temperature range spanned is less than 100 K wide. Nevertheless, now we have experimental data for the important Mg-sulfate hydrates kieserite (Frost et al., 1957), starkeyite (this study), hexahydrite (Cox et al., 1955), and epsomite (Gurevich et al., 2007). For starkeyite, the data reported in this study in the temperature region T=229–303 K (cf. Table 3a, 3b) were fitted with a commonly used Maier-Kelley polynomial \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*}C_p ( T ) = { \rm a + b}T + { \rm c}T^2 \tag{8}\end{align*}\end{document}

The fit parameters a, b, and c are listed in Table 4 together with the parameters for the other minerals, which were taken from Grevel and Majzlan (2009).

Table 4.

Maier-Kelley C _p Parameters for Several MgSO₄ Hydrates

	Kieserite	Starkeyite	Hexahydrite	Epsomite
a	103.124	107.925	203.1677	218.7748
b	0.1696	0.5532	0.6687	0.7465
c	−1794051	−1048894	−4466548	−4883464

The parameters for kieserite, hexahydrite, and epsomite are taken from Grevel and Majzlan (2009). The parameters for starkeyite result from a fit to the experimental data in the range 229 K<T<303 K.

In the present study, we repeated the MAP analysis carried out by Grevel and Majzlan (2009), suspending the assumption ΔC_p =0 for all reactions under consideration. The fugacity of steam is calculated by the equation of state of Grevel and Chatterjee (1992), which is based on the data of Haar et al. (1984) and is implemented in the software used for the MAP analysis (cf. Grevel and Majzlan, 2011). Standard state values for steam are taken from Robie and Hemingway (1995).

The inequalities given by the phase property constraints are defined by \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} \begin{split} \Delta_{ \rm f}H_{298 , { \rm observed}}^0 - 2 \sigma \big( \Delta_{ \rm f}H_{298 , { \rm observed}}^0 \big) < \Delta_{ \rm f} H_{298}^0 \\ < \Delta_{ \rm f} H_{298 , { \rm observed}}^0 + 2 \sigma \big( \Delta_{ \rm f} H_{298 , { \rm observed}}^0 \big)\end{split} \tag{9{\rm a}}\end{align*}\end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} \begin{split} S_{298 , { \rm observed}}^0 - 2 \sigma \big( S_{298, { \rm observed}}^0 \big) < S_{298}^0 \\ < S_{298 , { \rm observed}}^0 + 2 \sigma \big( S_{298 , { \rm observed}}^0 \big)\end{split} \tag{9{\rm b}}\end{align*}\end{document}

\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\Delta_{ \rm f}H_{298}^0$$\end{document} of the respective hydrates of MgSO₄ are treated as variables in the refinement procedure based on FORTRAN routines of the NAG FORTRAN-library (Numerical Algorithms Group Ltd., Oxford).

In a first run, the MAP problem was constrained by the same reversal experiments already used by Grevel and Majzlan (2009), which are depicted in Fig. 1 (van't Hoff et al., 1912; Chou and Seal, 2003, 2007). The calorimetric \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\Delta_{ \rm f}H_{298}^0$$\end{document} values for epsomite, hexahydrite, kieserite (Grevel and Majzlan, 2009), and starkeyite (this study) served as starting values; they were restricted to vary within their stated uncertainties. The standard entropies were taken from DeKock (1986) with the exception of epsomite (Gurevich et al., 2007) and starkeyite, where we used our new value. For the entropy, data variations of±1% were allowed during the MAP analysis.

The resulting values consistent both with our calorimetric data and humidity brackets are listed in Table 5a. The refined standard enthalpy values changed only slightly (generally less than 1 kJ·mol⁻¹) in comparison to the starting values, that is, well within their error limits given by the calorimetric measurements. On the other hand, the entropy values changed more significantly. The largest difference (2.9 J·mol⁻¹·K⁻¹) to the starting value 348.1 J·mol⁻¹·K⁻¹ was obtained for \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$S_{298}^0$$\end{document} (hexahydrite).

Table 5.

Standard Properties of Mg-Sulfate Hydrates Obtained by MAP Analysis

(a)
Phase	\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\Delta_f{ \rm H}_{298}^0$$\end{document} (kJ·mol⁻¹)	\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$Diff ( \Delta_f{ \rm H}_{298}^0)$$\end{document} (kJ·mol⁻¹)	\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$${ \rm S}_{298}^0$$\end{document} (J·K⁻¹·mol⁻¹)	\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$Diff ( { \rm S}_{298}^0)$$\end{document} (J·K⁻¹·mol⁻¹)	\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\Delta_f G_{298}^0$$\end{document} (kJ·mol⁻¹)
Epsomite	−3388.53	−0.834	373.82	−1.653	−2871.55
Hexahydrite	−3088.90	−0.804	345.20	−2.897	−2632.93
Starkeyite	−2497.71	0.988	256.76	2.281	−2154.46
Kieserite	−1612.10	0.399	126.04	−0.320	−1438.51

(b)
Phase	\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\Delta_f{ \rm H}_{298}^0$$\end{document} (kJ·mol⁻¹)	\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$Diff ( \Delta_f{ \rm H}_{298}^0)$$\end{document} (kJ·mol⁻¹)	\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$S_{298}^0$$\end{document} (J·K⁻¹·mol⁻¹)	\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$Diff ( { \rm S}_{298}^0)$$\end{document} (J·K⁻¹·mol⁻¹)	\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\Delta_f { \rm G}_{298}^0$$\end{document} (kJ·mol⁻¹)
Epsomite	−3388.11	−0.409	375.82	0.345	−2871.72
Hexahydrite	−3088.17	−0.066	348.25	0.145	−2633.10
Starkeyite	−2498.47	0.226	254.19	−0.290	−2154.45
Kieserite	−1612.33	0.170	126.22	−0.136	−1438.79

The differences \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\Delta_{ \rm f} H_{298 , { \rm observed}}^0 - \Delta_{ \rm f}H_{298 , { \rm refined}}^0$$\end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$S_{298 , { \rm observed}}^0 - S_{298 , { \rm refined}}^0$$\end{document} are given as well. The Gibbs free energy of formation data are based on the entropy data for S (orthorhombic), O₂, H₂, and Mg by Robie and Hemingway (1995).

(a) MAP problem constrained by reversals of Chou and Seal (2003, 2007), van't Hoff et al. (1912), and

(b) MAP problem constrained by reversals of Chou and Seal (2003), Steiger et al. (2011), van't Hoff et al. (1912).

Steiger et al. (2011) reported new measurements of equilibrium relative humidities for several stable and metastable hydration–dehydration equilibria in the MgSO₄–H₂O system (see Fig. 1). They concluded a good agreement of their data with those of Chou and Seal (2003) for the epsomite/hexahydrite reaction but a significant disagreement with the Chou and Seal (2007) experiments for the hexahydrite dehydration to starkeyite, especially at higher temperatures. Furthermore, the data of Steiger et al. (2011) are in a good agreement with the experimental data on the latter reaction provided by Lallemant et al. (1974). Additionally, Steiger et al. (2011) presented data for the reaction epsomite=kieserite + 6H₂O.

In a second run, the measurements of Steiger et al. (2011) served as constraints for our MAP procedure to obtain a revised internally consistent thermodynamic data set for magnesium sulfate hydrates. Since Steiger et al. (2011) did not provide brackets but only equilibrium positions, their stated overall accuracy (±1–2%RH) was added retrospectively to obtain brackets constraining the MAP problem. In this run, all starting values including the entropy values changed only slightly (cf. Table 5b).

Figure 6 shows the calculated equilibrium curves according to Eq. 4, for which the refined data were used, for different hydration/dehydration reactions. In Fig. 6a, the data are constrained by the brackets of Chou and Seal (2003, 2007), while in Fig. 6b the diagram constrained by the Steiger et al. (2011) data is shown. Obviously, the (metastable) dehydration hexahydrite/starkeyite is moved to higher temperatures here. The invariant point defined by MgSO₄·7H₂O + MgSO₄·6H₂O + MgSO₄·H₂O + vapor, which is equivalent to the lower temperature limit of the hexahydrite stability field, is located at about 7.5°C and 39.0%RH (Fig. 6a), respectively, 10.0°C and 40.7%RH (Fig. 6b). Both diagrams show calculated curves extended to temperatures below 0°C. The sublimation pressure of H₂O was calculated according to Wagner et al. (1994). The invariant point given by the assemblage MgSO₄·7H₂O +MgSO₄·6H₂O + MgSO₄·4H₂O + vapor, which was modeled at 3°C and 32.9%RH by Steiger et al. (2011), is now found below 0°C in both cases. When the MAP problem is constrained by the Chou and Seal (2003; 2007) data, this point is located at nearly −18°C and 27.2%RH, whereas constraining the data by the experiments of Steiger et al. (2011) yields about −13°C at 28.3%RH. Quantitatively, both diagrams are very similar to Fig. 6 of Chou and Seal (2007). At −50°C, the (metastable) dehydration of epsomite to starkeyite occurs at about 20%RH and the epsomite–kieserite reaction at ca. 25%RH; this behavior is almost similar in Fig. 6a and 6b and slightly different from that of Steiger et al. (2011), who calculate the equilibrium humidities at this temperature to about 12%RH respectively 15%RH. It has to be noted here that Steiger et al. (2011) based their calculations on a pure water standard state at subzero temperatures, that is, RH=p/p⁰, with p⁰ the vapor pressure over supercooled liquid water. If the sublimation pressure, that is, the saturation vapor pressure over ice, represented by the ice-line in their diagram, is used, the calculated equilibrium humidities at T=−50°C are 21%RH for the epsomite–starkeyite transition and 25.6%RH for the epsomite–kieserite transition. So actually there is excellent agreement with their calculations at low temperature.

FIG. 6.

Calculated stability of Mg-sulfate hydrates as a function of temperature and relative humidity. Solid lines represent the stable equilibria MgSO₄·7H₂O⇔MgSO₄·6H₂O + H₂O (epsomite/hexahydrite), MgSO₄·6H₂O⇔MgSO₄·H₂O + 5H₂O (hexahydrite/kieserite), and MgSO₄·7H₂O⇔MgSO₄·H₂O + 6H₂O (epsomite/kieserite below 10°C); dashed curves are the metastable MgSO₄·6H₂O⇔MgSO₄·4H₂O + 2H₂O (hexahydrite/starkeyite), MgSO₄·7H₂O⇔MgSO₄·4H₂O + 3H₂O (epsomite/starkeyite below −13°C), and MgSO₄·7H₂O⇔MgSO₄·6H₂O + H₂O (epsomite/hexahydrite below 10°C) coexistence curves; the dotted line represents the MgSO₄·7H₂O⇔MgSO₄·H₂O + 6H₂O boundary (epsomite/kieserite above 10°C). The solubility curve was calculated after Steiger et al. (2011). Large symbols represent experimentally obtained reversals of Chou and Seal (2003), Chou and Seal (2007), and van't Hoff et al. (1912). Small symbols represent measurements of equilibrium relative humidities for several hydration–dehydration equilibria in the MgSO₄–H₂O system (Steiger et al., 2011). The meridianiite (n=11)–epsomite reaction is not shown in the diagrams because no experimentally determined thermodynamic properties for meridianiite are available.

6. Implications

As already pointed out by Chou and Seal (2007) and Steiger et al. (2011), magnesium sulfate hydrates may play a dominant role in the water cycle of Mars through hydration and dehydration reactions involving atmospheric moisture during diurnal cycles of heating and cooling. Meridianiite (n=11) and epsomite (n=7) are stable at higher relative humidities during the night, while kieserite (n=1) is the stable phase during the martian day conditions. Starkeyite (n=4) is metastable in the entire T-%RH range. Kieserite and higher hydrated phases, so-called “polyhydrated sulfates,” are undoubtedly present on Mars, but the exact nature of the latter phases is not known up to now. Possible candidates include higher hydrated magnesium sulfates but also hydrated sulfates with other cations and double compounds such as bloedite (Roach et al., 2009). Strong kinetic limitations on Mg-sulfate equilibrium may prevent kieserite hydration (Vaniman and Chipera, 2006). Also, dehydration to kieserite is considered to be very sluggish. Therefore, the metastable formation of starkeyite is also possible under martian conditions. Though favorable from a thermodynamic point of view, the low temperatures on Mars might slow down reaction rates such that daily cycling is too short to allow for significant changes in the state of hydration. This may explain the occurrence of both kieserite and higher hydrated sulfates.

Interpretation of the potential role of magnesium sulfate hydrates in the martian water cycle must be approached with caution because of the uncertainties in reaction pathways at the low temperatures of the martian surface, which have not yet been investigated experimentally in enough detail. Further experimental data are needed at subfreezing temperatures to bracket Mg-sulfate stability. Nevertheless, this work places constraints on the conditions for preservation of the metastable tetra- and trihydrates of MgSO₄, which is an essential requirement for interpreting the possible detection of these species by the forthcoming Curiosity Mars rover's mineralogy package, which includes in situ X-ray diffraction.

7. Concluding Remarks

In this work, we have derived two slightly different sets of internally consistent thermodynamic data for several MgSO₄ hydrates by a mathematical programming procedure, including data for starkeyite based on calorimetric measurements. These sets are based on the new equilibrium experiments of Steiger et al. (2011) on one hand and on the brackets provided by Chou and Seal (2003, 2007) on the other (Fig. 1). The deviations from the input values are significantly smaller when the MAP problem is constrained by the Steiger et al. (2011) data, which are, on the other hand, more scattered than the data by Chou and Seal (2003, 2007). Since the treatment of the hexahydrite–starkeyite equilibrium by Steiger et al. (2011) is also consistent with all reported solubilities of the two solids (Steiger, personal communication), we prefer the use of the data listed in Table 5b. Nevertheless, both data sets are internally consistent. To constrain the data better, more low temperature calorimetric data are needed.

Footnotes

Acknowledgments

This work was supported by the Deutsche Forschungsgemeinschaft (Grant MA 3927/5-1 and 5-2), which is highly appreciated. Also, grants of the Austrian Science Fund (FWF), project numbers P21370 to E. Dachs and P23056 to A. Benisek, are gratefully acknowledged.

Abbreviations

DSC, differential scanning calorimetry, differential scanning calorimeter; DTA, differential thermal analysis; MAP, mathematical programming; PPMS, physical property measurement system; TG, thermogravimetric analysis; XRD, X-ray powder diffraction.

1

The reference van't Hoff et al. (1912) of Grevel and Majzlan (2009) is equivalent to van't Hoff et al. (1901) given by Steiger et al. ().

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