An equation for the heat capacity of solids

A new expression for the heat capacity of solids is proposed: a) for fitting the measured low-temperature heat capacity data on crystals, and b) for estimating heat capacity at high temperatures without violating the limit set by Petit and Dulong (1819). C_p = 3Rn(1 + k₁T⁻¹ + k₂T⁻² + k₃T⁻³) + (A + BT) + C'_p where R and n are the gas constant and the number of atoms in the chemical formulae, respectively. A and B are calculated from thermal expansion coefficient and isothermal bulk modulus data. The k_i's are determined by fitting the measured low-temperature heat capacity data. C'_p is the departures from the 3Rn limit for some substances due to cation disordering, anharmonicity and electronic contributions.The five constants in the proposed equation were determined for six minerals for which the physical properties have been well studied. The errors possibly resulting from experimental measurements or extrapolations of the thermal expansion coefficient and bulk modulus were carefully examined.     

A high P-T single-crystal X-ray diffraction study of thermoelasticity of MgSiO3 orthoenstatite

P-V-T data of MgSiO₃ orthoenstatite have been measured by single-crystal X-ray diffraction at simultaneous high pressures (in excess of 4.5 GPa) and temperatures (up to 1000 K). The new P-V-T data of the orthoenstatite, together with previous compression data and thermal expansion data, are described by a modified Birch-Murnaghan equation of state for diverse temperatures. The fitted thermoelastic parameters for MgSiO₃ orthoenstatite are: thermal expansion ?α/?P with values of a=2.86(29)×10^-5 K^-1 and b=0.72(16)×10^-8 K^-2; isothermal bulk modulus K _{T o} =102.8(2) GPa; pressure derivative of bulk modulus K′=?K/?P=10.2(1.2); and temperature derivative of bulk modulus K=?K/?T=-0.037(5) GPa/K. The derived thermal Grüneisen parameter is γ _th=1.05 for ambient conditions; Anderson-Grüneisen parameter is δ _{T o} =11.6, and the pressure derivative of thermal expansion is ?α/?P=-3.5×10^-6K^-1 GPa^-1. From the P-V-T data and the thermoelastic equation of state, thermal expansions at two constant pressures of 1.5 GPa and 4.0 GPa are calculated. The resulting pressure dependence of thermal expansion is Δα/ΔP=-3.2(1)× 10^-6 K^-1 GPa^-1. The significantly large values of K′, K, δ _T and ?α/?P indicate that compression/expansion of MgSiO₃ orthoenstatite is very sensitive to changes of pressure and temperature.     

Raman spectroscopy and heat capacity measurement of calcium ferrite type MgAl2O4 and CaAl2O4

Raman spectroscopy of calcium ferrite type MgAl₂O₄ and CaAl₂O₄ and heat capacity measurement of CaAl₂O₄ calcium ferrite were performed. The heat-capacity of CaAl₂O₄ calcium ferrite measured by a differential scanning calorimeter (DSC) was represented as C_P(T)=190.6–1.116 × 10⁷T^–2 + 1.491 × 10⁹T^–3 above 250 K (T in K). The obtained Raman spectra were applied to lattice dynamics calculation of heat capacity using the Kieffer model. The calculated heat capacity for CaAl₂O₄ calcium ferrite showed good agreement with that by the DSC measurement. A Kieffer model calculation for MgAl₂O₄ calcium ferrite similar to that for CaAl₂O₄ calcium ferrite was made to estimate the heat capacity of the former. The heat capacity of MgAl₂O₄ calcium ferrite was represented as C_P(T)=223.4–1352T ^–0.5 – 4.181 × 10⁶T ^–2 + 4.300 × 10⁸T ^–3 above 250 K. The calculation also gave approximated vibrational entropies at 298 K of calcium ferrite type MgAl₂O₄ and CaAl₂O₄ as 97.6 and 114.9 J mol^–1 K^–1, respectively.     

Anharmonicity and high-temperature heat capacity of crystals: the examples of Ca2GeO4, Mg2GeO4 and CaMgGeO4 olivines

Drop-calorimetry determinations of the isobaric heat capacity (C_P) of Mg₂GeO₄, Ca₂GeO₄ and CaMgGeO₄ have been made up to 1700 K. The thermal expansion coefficient (α) of these olivine germanates has been determined from high-temperature X-ray measurements up to 1500 K. From these measurements and available compressibility data, one calculates that the isochoric heat capacity (C_V) exceeds the harmonic limit of Dulong and Petit above 1000–1200 K. Such an intrinsic anharmonic behaviour can be accounted for by introducing anharmonic parameters a_i=(? ln v _i/?T)_V in vibrational modelling of C_V. These parameters are calculated from pressure and temperature shifts of the vibrational frequencies as measured by Raman spectroscopy up to 10 GPa at room temperature and up to 1300 K at 1 bar. A comparison of the Raman spectra of the three germanates with those of natural olivines justifies once again the use of germanates as silicate analogues. Extensive Ca,Mg disordering likely takes place in CaMgGeO₄, beginning at about 1100 K and leading to unusually high increases of the heat capacity and thermal expansion coefficient.     

A calorimetric investigation of spessartine: Vibrational and magnetic heat capacity

The heat capacity (C_p) 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 C_p polynomial for T > 250 K:

     

Inverted high-temperature quartz

Fifty-two samples of inverted high-temperature quartz from volcanic rocks were investigated by Guinier-Jago powder diffractometry and differential scanning calorimetry (DSC). Quartz megacrysts from Clear Lake and Cinder Cone, California show a variability of ?2.5 ° K in their α-β transition temperature (T _α-β). Quartz phenocrysts and quartz from crystalline rocks give a range of 0.5 ° K in T _α-β. Neutron activation analysis of single crystals demonstrates that Al is the principal impurity (17–380 ppm). Its concentration is inversely correlated with T _α-β. A very small variation was found in the a and c lattice parameters among the specimens of volcanic quartz studied. This variation does not correlate with Al content or transition temperature. Mean values at 22 ° C (a=4.1934±0.0004 Å, c=5.4046±0.0006 Å) are similar to those of quartz grown at low temperatures. Enthalpy of the α-β transition (ΔH _α-β), obtained over 9.0 ° from DSC runs, is dependent upon sample grain size and for a crushed powder with zero hysteresis (T _α-β on heating=T _α-β on cooling) is 92.0 ±1.4 cal/mol. In contrast, a single piece of quartz requires ΔH _α-β be 107.7±1.4 cal/mol and has a T _α-β hysteresis of 1.1 ° K. Regression of published data provides equations for the variation of the molar volume (cc/mol) of quartz with v. These equations imply a ΔV _α-β of 0.205±0.031 cc/- mol. Expressions are also provided for the temperature dependence of the thermal coefficient of expansion, α, the compressibility, β, and (?/gb/?T)_p (which is identically -(?α/?P) _T ). DSC heat capacity measurements over the range 400 to 900 ° K were fitted to extended Maier-Kelley type expressions to give: $$\begin{gathered} C_P = 10.31 + 9.116 \times 10^{ - 3} T - \frac{{1.812 \times 10^5 }}{{T^2 }} \hfill \\ - {\text{5}}{\text{.630}} \times 10^{ - 2} {\text{ }}\frac{T}{{(T - 848)}} - 0.3553\frac{T}{{(T - 848)^2 }} \hfill \\ - 0.9011\frac{T}{{\left( {T - 848} \right)^3 }} \hfill \\ (400{\text{ to 842}}^ \circ {\text{K), and}} \hfill \\ C_P = - 318.8 + 0.2532T \hfill \\ {\text{ + }}\frac{{8.687 \times 10^7 }}{{T^2 }} + 0.1603\frac{T}{{\left( {T - 848} \right)^4 }} \hfill \\ \end{gathered} $$ (851 to 900 ° K), which together with the values of ΔH _α?β measured over the range 842–851° K give 7875.3 cal/mol for H₉₀₀-H₄₀₀. The behavior of α, β, and C _p as a function of T emphasizes that structural changes which occur at the α?β transition do so over a broad temperature interval.     

The high-temperature heat capacity of natural calcite (CaCO3)

The heat capacity (C _P) of a natural sample of calcite (CaCO₃) has been measured from 350 to 775 K by differential scanning calorimetry (DSC). Heat capacities determined for a powdered sample and a single-crystal disc are in close agreement and have a total uncertainty of ±1 percent. The following equation for the heat capacity of calcite from 298 to 775 K was fit by least squares to the experimental data and constrained to join smoothly with the low-temperature heat capacity data of Staveley and Linford (1969) (C _P in J mol^?1 K^?1, T in K): $$\begin{gathered} C_p = - 184.79 + 0.32322T - 3,688,200T^{ - 2} \hfill \\ {\text{ }} - (1.2974{\text{ }} \times {\text{ 10}}^{ - {\text{4}}} )T^2 + 3,883.5T^{ - 1/2} \hfill \\ \end{gathered} $$ Combining this equation with the S ₂₉₈ ⁰ value from Staveley and Linford (1969), entropies for calcite are calculated and presented to 775 K. A simple method of extrapolating the heat capacity function of calcite above 775 K is presented. This method provides accurate entropies of calcite for high-temperature thermodynamic calculations, as evidenced by calculation of the equilibrium: CaCO_{3 (s)}=CaO_(s)+CO_{2 (s)}.     

Thermal expansion and crystal structure of FeSi between 4 and 1173 K determined by time-of-flight neutron powder diffraction

The thermal expansion and crystal structure of FeSi has been determined by neutron powder diffraction between 4 and 1173?K. No evidence was seen of any structural or magnetic transitions at low temperatures. The average volumetric thermal expansion coefficient above room temperature was found to be 4.85(5)?×?10^?5?K^?1. The cell volume was fitted over the complete temperature range using Grüneisen approximations to the zero pressure equation of state, with the internal energy calculated via a Debye model; a Grüneisen second-order approximation gave the following parameters: θ_D=445(11)?K, V ₀=89.596(8)?ų, K ₀′=4.4(4) and γ′=2.33(3), where θ_D is the Debye temperature, V ₀ is V at T=0?K, K ₀′ is the first derivative with respect to pressure of the incompressibility and γ′ is a Grüneisen parameter. The thermodynamic Grüneisen parameter, γ_th, has been calculated from experimental data in the range 4–400?K. The crystal structure was found to be almost invariant with temperature. The thermal vibrations of the Fe atoms are almost isotropic at all temperatures; those of the Si atoms become more anisotropic as the temperature increases.     

Effects of temperature on the crystal structure of epidote: a neutron single-crystal diffraction study at 293 and 1,070 K

The effects of temperature on the crystal structure of a natural epidote [Ca_1.925 Fe_0.745Al_2.265Ti_0.004Si_3.037O₁₂(OH), a = 8.890(6), b = 5.630(4), c = 10.150(6) Å and β = 115.36(5)°, Sp. Gr. P2₁ /m] have been investigated by means of neutron single-crystal diffraction at 293 and 1,070 K. At room conditions, the structural refinement confirms the presence of Fe³⁺ at the M₃ site [%Fe(M3) = 73.1(8)%] and all attempts to refine the amount of Fe at the M(1) site were unsuccessful. Only one independent proton site was located. Two possible hydrogen bonds, with O(2) and O(4) as acceptors [i.e. O(10)–H(1)···O(2) and O(10)–H(1)···O(4)], occur. However, the topological configuration of the bonds suggests that the O(10)–H(1)···O(4) is energetically more favourable, as H(1)···O(4) = 1.9731(28) Å, O(10)···O(4) = 2.9318(22) Å and O(10)–H(1)···O4 = 166.7(2)°, whereas H(1)···O(2) = 2.5921(23) Å, O(10)···O(2) = 2.8221(17) Å and O(10)–H(1)···O2 = 93.3(1)°. The O(10)–H(1) bond distance corrected for “riding motion” is 0.9943 Å. The diffraction data at 1,070 K show that epidote is stable within the T-range investigated, and that its crystallinity is maintained. A positive thermal expansion is observed along all the three crystallographic axes. At 1,070 K the structural refinement again shows that Fe³⁺ share the M(3) site along with Al³⁺ [%Fe(M3)_1,070K = 74(2)%]. The refined amount of Fe³⁺ at the M(1) is not significant [%Fe(M1)_1,070K = 1(2)%]. The tetrahedral and octahedral bond distances and angles show a slight distortion of the polyhedra at high-T, but a significant increase of the bond distances compared to those at room temperature is observed, especially for bond distances corrected for “rigid body motions”. The high-T conditions also affect the inter-polyhedral configurations: the bridging angle Si(2)–O(9)–Si(1) of the Si₂O₇ group increases significantly with T. The high-T structure refinement shows that no dehydration effect occurs at least within the T-range investigated. The configuration of the H-bonding is basically maintained with temperature. However, the hydrogen bond strength changes at 1,070 K, as the O(10)···O(4) and H(1)···O(4) distances are slightly longer than those at 293 K. The anisotropic displacement parameters of the proton site are significantly larger than those at room condition. Reasons for the thermal stability of epidote up to 1,070 K observed in this study, the absence of dehydration and/or non-convergent ordering of Al and Fe³⁺ between different octahedral sites and/or convergent ordering on M(3) are discussed.     

Heat capacity and thermodynamic properties of andradite garnet,Ca3Fe2Si3O12, between 10 and 1000 K and revised values for ΔfGom (298.15 K) of hedenbergite and wollastonite

The heat capacity of synthetic andradite garnet (Ca₃Fe₂Si₃O₁₂) was measured between 9.6 and 365.5 K by cryogenic adiabatic calorimetry and from 340 to 990 K by differential scanning calorimetry. At 298.15 K C^o_p,m and S^o_m are 351.9 ± 0.7 and 316.4 ± 2.0 J/(mol·K), respectively.Andradite has a λ-peak in C^o_p,m with a maximum at 11.7 ± 0.2 K which is presumably associated with the antiferromagnetic ordering of the magnetic moments of the Fe³⁺ ions. The Gibbs free energy of formation, Δ_fG^o_m (298.15 K) of andradite is −5414.8 ± 5.5 kJ/mol and was obtained by combining our entropy and heat capacity data with the known breakdown of andradite to pseudowollastonite and hematite at ≈ 1410 to 1438 K. From a reexamination of the calcite + quartz = wollastonite equilibrium data we obtained Δ_fH^o_m (298.15 K) = − 1634.5 ± 1.8 kJ/mol for wollastonite.Between 300 and 1000 K the molar heat capacity of andradite can be represented by the equation C^o_p,m = 809.24 - 7.025 × 10−2T− 7.403 × 10³T−0.5 − 6.789 × 10⁵T−2. We have also used our thermochemical data for andradite to estimate the Gibbs free energy of formation of hedenbergite (CaFeSi₂O₆) for which we obtained Δ_fG^o_m (298.15 K) = −2674.3 ± 5.8 kJ/mol.     

Stability and P–V–T equation of state of KAlSi3O8-hollandite determined by in situ X-ray observations and implications for dynamics of subducted continental crust material

In situ X-ray diffraction measurements of KAlSi₃O₈-hollandite (K-hollandite) were performed at pressures of 15–27 GPa and temperatures of 300–1,800 K using a Kawai-type apparatus. Unit-cell volumes obtained at various pressure and temperature conditions in a series of measurements were fitted to the high-temperature Birch-Murnaghan equation of state and a complete set of thermoelastic parameters was obtained with an assumed K′_300,0=4. The determined parameters are V _300,0=237.6(2) Å³, K _300,0=183(3) GPa, (?K _T,0/?T) _P =?0.033(2) GPa K^?1, a ₀=3.32(5)×10^?5 K^?1, and b ₀=1.09(1)×10^?8 K^?2, where a ₀ and b ₀ are coefficients describing the zero-pressure thermal expansion: α _T,0 = a ₀ + b ₀ T. We observed broadening and splitting of diffraction peaks of K-hollandite at pressures of 20–23 GPa and temperatures of 300–1,000 K. We attribute this to the phase transitions from hollandite to hollandite II that is an unquenchable high-pressure phase recently found. We determined the phase boundary to be P (GPa)=16.6 + 0.007 T (K). Using the equation of state parameters of K-hollandite determined in the present study, we calculated a density profile of a hypothetical continental crust (HCC), which consists only of K-hollandite, majorite garnet, and stishovite with 1:1:1 ratio in volume. Density of HCC is higher than the surrounding mantle by about 0.2 g cm^?3 in the mantle transition zone while this relation is reversed below 660-km depth and HCC becomes less dense than the surrounding mantle by about 0.15 g cm^?3 in the uppermost lower mantle. Thus the 660-km seismic discontinuity can be a barrier to prevent the transportation of subducted continental crust materials to the lower mantle and the subducted continental crust may reside at the bottom of the mantle transition zone.     

Mössbauer effect study of anapaite, Ca2Fe2+(PO4)2 · 4H2O, and of its oxidation products

Mössbauer spectra (MS) of anapaite (Ca₂ Fe²⁺(PO₄)₂?·?4H₂O) and of a sample after being immersed in a 4% H₂O₂ solution at room temperature (RT) over 12 days (hereafter an4ox) were collected at temperatures in the range 4.2 to 420?K and 11 to 300?K respectively. All MS consist of symmetrical doublets, hence magnetic ordering was not observed. The temperature dependencies of the Fe²⁺ centre shifts of anapaite and an4ox were analysed with the Debye model for the lattice vibrations. The characteristic Mössbauer temperatures were found as 370?K?±?25?K and 340?K?±?25?K, and the intrinsic isomer shifts as 1.427?±?0.005?mm/s and 1.418?±?0.005?mm/s respectively. From the external-field (60?kOe) MS recorded at 4.2 and 189?K for the non-treated sample, the principal component V _zz of the electric field gradient (EFG) is determined to be positive and the asymmetry parameter η?≈?0.2 and 0.4 respectively. The temperature variations of the quadrupole splittings, ΔE _Q(T), cannot be interpreted on the basis of the thermal population of the ⁵ D electronic levels resulting from the tetragonal compression of the O₆ co-ordination. The low-temperature linear behaviour of ΔE _Q(T) is attributed to a strong orbit-lattice coupling. A field of 60 kOe applied to anapaite at 4.2?K produces magnetic hyperfine splitting with effective hyperfine fields of ?136, ?254 and ?171?kOe along the principal axes Ox, Oy and Oz of the EFG tensor respectively. Additional oxidation treatments in solutions with various H₂O₂ concentrations up to 20% and subsequent Mössbauer experiments at room temperature, have revealed that the anapaite structure is not sensitive to oxidation since eventually only a small amount of Fe²⁺ (～6.5%) is converted into Fe³⁺.     

High-temperature thermal expansion of lime, periclase, corundum and spinel

The high-temperature cell parameters of lime (CaO), periclase (MgO), corundum (Al₂O₃), and spinel (MgAl₂O₄) have been determined from 300 up to 3000 K through X-ray diffraction experiments with synchrotron radiation. The good agreement found with dilatometric results suggests that vacancy-type defects do not make a large contribution to thermal expansion for these oxides, even near the melting point, justifying the use of X-ray diffraction for determining volume properties up to very high temperatures. Thermal expansion coefficients were determined from the measured cell volumes with equations of the form α=α₀ + α₁ T + α₂/T ². Along with available isobaric heat capacity and compressibility data, these derived coefficients clearly show that anharmonic effects contribute little to the isochoric heat capacities (C _v ) of CaO, MgO, and Al₂O₃, which do not depart appreciably from the 3nR Dulong and Petit limit. Received: 31 March 1999 / Revised, accepted: 23 June 1999     

Thermodynamic properties of copper carbonates — malachite Cu2(OH)2CO3 and azurite Cu3(OH)2(CO3)2

The thermodynamic properties of the copper carbonates malachite and azurite have been studied by adiabatic calorimetry, by heat-flux Calvet Calorimetry, by differential thermal analysis (DTA) and by thermogravimetrie (TGA) analysis. The heat capacities, C _p ⁰ of natural malachite and azurite have been measured between 3.8 and 300 K by low-temperature adiabatic calorimetry. The heat capacity of azurite exhibits anomalous behavior at low temperatures. At 298.15 K the molar heat capacities C _p ⁰ and the third law entropies S _298.15 ⁰ are 228.5±1.4 and 254.4±3.8 J mol^?1 K^?1 for azurite and 154.3±0.93 and 166.3±2.5 J mol^?1 K^?1 for malachite. Enthalpies of solution at 973 K in lead borate 2PbO·B₂O₃ have been measured for heat treated malachite and azurite. The enthalpies of decomposition are 105.1±5.8 for azurite and 66.1±5.0 kJ mol^? for malachite. The enthalpies of formation from oxides of azurite and malachite determined by oxide melt solution calorimetry, are ?84.7±7.4 and ?52.5±5.9 kJ mol^?1, respectively. On the basis of the thermodynamic data obtained, phase relations of azurite and malachite in the system Cu²⁺-H₂O-CO₂ at 25 and 75 °C have been studied.     

Crystal structure, Raman and FTIR spectroscopy, and equations of state of OH-bearing MgSiO3 akimotoite

MgSiO₃ akimotoite is stable relative to majorite-garnet under low-temperature geotherms within steeply or rapidly subducting slabs. Two compositions of Mg–akimotoite were synthesized under similar conditions: Z674 (containing about 550 ppm wt H₂O) was synthesized at 22 GPa and 1,500 °C and SH1101 (nominally anhydrous) was synthesized at 22 GPa and 1,250 °C. Crystal structures of both samples differ significantly from previous studies to give slightly smaller Si sites and larger Mg sites. The bulk thermal expansion coefficients of Z674 are (153–839 K) of a ₁ = 20(3) × 10^?9 K^?2 and a ₀ = 17(2) × 10^?6 K^?1, with an average of α ₀ = 27.1(6) × 10^?6 K^?1. Compressibility at ambient temperature of Z674 was measured up to 34.6 GPa at Sector 13 (GSECARS) at Advanced Photon Source Argonne National Laboratory. The second-order Birch–Murnaghan equation of state (BM2 EoS) fitting yields: V ₀ = 263.7(2) Å³, K _T0 = 217(3) GPa (K′ fixed at 4). The anisotropies of axial thermal expansivities and compressibilities are similar: α _a  = 8.2(3) and α _c  = 10.68(9) (10^?6 K^?1); β _a  = 11.4(3) and β _c  = 15.9(3) (10^?4 GPa). Hydration increases both the bulk thermal expansivity and compressibility, but decreases the anisotropy of structural expansion and compression. Complementary Raman and Fourier transform infrared (FTIR) spectroscopy shows multiple structural hydration sites. Low-temperature and high-pressure FTIR spectroscopy (15–300 K and 0–28 GPa) confirms that the multiple sites are structurally unique, with zero-pressure intrinsic anharmonic mode parameters between ?1.02 × 10^?5 and +1.7 × 10^?5 K^?1, indicating both weak hydrogen bonds (O–H···O) and strong OH bonding due to long O···O distances.     

The partial molar volume, thermal expansivity, and compressibility of H2O in NaAlSi3O8 liquid: new measurements and an internally consistent model

The molar volumes of 19 hydrous albitic liquids (1.9 to 6.1 wt% H₂O^total) were determined at one bar and 505–765 K. These volume data were derived from density measurements on hydrous glasses at 298 K, followed by measurements of the thermal expansion of each glass from 298 K to its respective glass transition temperature. The technique exploits the fact that the volume of a glass is equal to that of the corresponding liquid at the limiting fictive temperature (T _f′), and that T _f′ can be approximated as the temperature near the onset of the rapid increase in thermal expansion that occurs in the glass transition interval. The volume data of this study were combined with available volume data for anhydrous, Na₂O-Al₂O₃-SiO₂ liquids to derive the partial molar volume (±1) of the H₂O component in an albitic melt at ∼565 K and one bar. To extend the determination of to higher temperatures and pressures, the molar volumes of the hydrous albitic liquids determined in this study were combined with those measured by previous authors at 1023–1223 K and 480–840 MPa, leading to the following fitted values (±1) at 1673 K and one bar: (±0.46)×10⁻³ cm⁻³/mol-K, and dVˉ ^H ₂ ^{O total} /dP=−3.82 (±0.36)×10⁻⁴ cm³/mol-bar. The measured molar volumes of this study and those of previous authors can be recovered with a standard deviation of 0.5%, which is within the respective experimental errors. There is a significant difference between the values for derived in this study as a function of temperature and pressure and those obtained from an existing polynomial, primarily caused by the previous absence of accurate density measurements on anhydrous silicate liquids. The coefficients of thermal expansion (=4.72×10⁻⁴/K) and isothermal compressibility ( _T =1.66×10⁻⁵/bar) for the H₂O component at 1273 K and 100 MPa, indicate that H₂O^total is the single most expansive and compressible component in silicate liquids. For example, at 1473 K and 70 MPa (conditions of a mid-ocean ridge crustal magma chamber), the presence of just 0.4 wt% H₂O will decrease the density of a basaltic liquid by more than one percent. An equivalent decrease in melt density could be achieved by increasing the temperature by 175 degrees or the decreasing pressure by 230 MPa. Therefore, even minor quantities of dissolved water will have a marked effect on the dynamic properties of silicate liquids in the crustal environment. Received: 20 August 1996 / Accepted: 15 March 1997     

The thermodynamic properties and phase relations of some minerals in the system CaO-AI2O3-SiO2-H2O

The heat capacities of lawsonite, margante, prehnite and zoisite have been measured from 5 to 350 K with an adiabatic-shield calorimeter and from 320 to 999.9 K with a differential-scanning calorimeter. At 298.15 K, their heat capacities, corrected to end-member compositions, are 66.35, 77.30, 79.13 and 83.84 cal K^?1 mol^?1; their entropies are 54.98, 63.01, 69.97 and 70.71 cal K^?1 mol^?1, respectively. Their high-temperature heat capacities are described by the following equations (in calories, K, mol): Lawsonite (298–600 K): Cp° = 66.28 + 55.95 × 10^?3T ? 15.27 × 10⁵T^?2 Margarite (298–1000 K): Cp° = 101.83 + 24.17 × 10^?3T ? 30.24 × 10⁵T^?2 Prehnite (298–800 K): Cp° = 97.04 + 29.99 × 10^?3T ? 25.02 × 10⁵T^?2 Zoisite (298–730 K): Cp° = 98.92 + 36.36 × 10^?3T ? 24.08 × 10⁵T^?2 Calculated Clapeyron slopes for univariant equilibria in the CaO-Al₂O₃-SiO₂-H₂O system compare well with experimental results in most cases. However, the reaction zoisite + quartz = anorthite + grossular + H₂O and some reactions involving prehnite or margarite show disagreements between the experimentally determined and the calculated slopes which may possibly be due to disorder in experimental run products. A phase diagram, calculated from the measured thermodynamic values in conjunction with selected experimental results places strict limits on the stabilities of prehnite and assemblages such as prehnite + aragonite, grossular + lawsonite, grossular + quartz, zoisite + quartz, and zoisite + kyanite + quartz. The presence of this last assemblage in eclogites indicates that they were formed at moderate to high water pressure.     

Heat capacity of wadeite-type K2Si4O9 and the pressure-induced stable decomposition of K-feldspar

The heat capacity of synthetic, stoichiometric wadeite-type K₂Si₄O₉ has been measured by DSC in the 195≤T(K)≤598 range. Near the upper temperature limit of our data, the heat capacity observed by DSC agrees with that reported by Geisinger et al. (1987) based on a vibrational model of their infrared and Raman spectroscopic data. However, with decreasing temperature, the Cp observed by DSC is progressively higher than that predicted from the vibrational model, suggesting that the standard entropy of K₂Si₄O₉ is likely to be larger than 198.9 ± 4.0 J/K · mol computed from the spectroscopic data. A fit to the DSC data gave: Cp(T) = 499.13 (±1.87) − 4.35014 · 10³(±3.489 · 10¹) · T ^−0.5, with T in K and average absolute percent deviation of 0.37%. The room-temperature compressibilities of kalsilite and leucite, hitherto unknown, have been measured as well. The data, fitted to the Murnaghan equation of state, gave K _o = 58.6 GPa, K _o ^′ = 0.1 for kalsilite and K _o = 45 GPa, K _o ^′ = 5.7 for α-leucite. Apart from the above mentioned data on the properties of the individual phases, we have also obtained reaction-reversals on four equilibria in the system K₂O-Al₂O₃-SiO₂. The Bayesian method has been used simultaneously to process the properties of 13 phases and 15 reactions between them to derive an internally consistent thermodynamic dataset for the K₂O-Al₂O₃-SiO₂ ternary. The enthalpy of formation of K₂Si₄O₉ wadeite is in perfect agreement with its revised calorimetric value, the standard entropy is 232.1 ± 10.4 J/K · mol, ∼15% higher than that implied by vibrational modeling. The phase diagram, generated from our internally consistent thermodynamic dataset, shows that for all probable P-T trajectories in the subduction regime, the stable pressure-induced decomposition of K-feldspar will produce coesite + kalsilite rather than coesite + kyanite + K₂Si₄O₉ (cf. Urakawa et al. 1994). Received: 11 June 1997 / Accepted: 2 December 1997     

Thermal diffusivity of garnets at high temperature

Thermal diffusivity (D) of garnets with diverse chemical compositions was measured using the laser-flash technique, which is accurate (±2%) and isolates the lattice component from direct radiative transfer. Temperatures ranged from ~290 to ~1,600 K (unless limited by melting). Seven synthetic (e.g., YAG, GGG) and 15 natural garnets with two types of ionic substitution [Ca₃(Fe,Al)₂Si₃O₁₂ and (Mg,Fe,Ca)₃Al₂Si₃O₁₂] and varying amounts of OH^- were examined. Cation substitution or hydroxyl incorporation lowers D from end-member values. Thermal diffusivity is constant once the temperature (T) exceeds a critical value (T _sat) of ~1,100 to 1,500 K. From ~290 K to T _sat, the measurements are best represented by 1/D=A+BT+CT ² where A, B, and C are constants. These constants vary little among diverse chemical compositions, suggesting that the oxygen sublattice controls heat transport. Higher order terms are needed only when T _sat is low, such as Ant Hill garnet wherein 1/D=0.049403+0.0032299T−2.3992T ²×10⁻⁶+6.0168T ³×10⁻¹⁰(1/D in s/mm²; T in K). The mean free path (λ, computed from D and sound velocities) is slightly larger than the lattice parameter above T _sat, in accord with phonon–phonon interactions requiring non-localized modes. At most temperatures, λ is nm-sized. Large values of λ are obtained by extrapolation to a few Kelvins, suggesting that boundary scattering can only be important at extremely cold temperatures. The observed behavior with T and chemical composition is consistent with the damped harmonic oscillator model. Phonon transport is best represented by inverse thermal diffusivity wherein 1/D goes as T ⁿ where n is between 1 and 3 up to ~200 K, depends on a quadratic or cubic polynomial at moderate T, but is constant above T _sat. The predicted and observed temperature response of 1/D mimics the well-known form for heat capacity, in that acoustic modes control heat transport near cryogenic temperatures, optic phonons dominate above ambient temperature, and a limit analogous to that of Dulong and Petit is reached at very high temperature, due to full population of discrete phonon states.     

Measurement of elastic properties of single-crystal CaO up to 1200 K

The elastic moduli of a single-crystal calcium oxide, CaO, are measured in the temperature range from 300 to 1200 K (1.8 times of the Debye temperature) by the resonant sphere technique (RST). The lowest 18 modes are identified in the frequency range from 0.6 to 1.4 MHz for the vibrating spherical specimen, which is 5.6564 mm in diameter and 3.3493 g/cm³ in density at room temperature, and the resonant frequencies are traced as a function of temperature. The adiabatic elastic moduli are determined in the present temperature range from the observed frequencies by inversion calculations. Most of the elastic moduli, except forC ₁₂ modulus, decrease as temperature increases. The temperature curves ofC _s andC ₄₄ moduli cross at 372 K. This means that the CaO specimen has an isotropic elasticity at the temperature. The temperature derivatives (?C ₁₁/?T) _P and (?C _s/?T) _P become slightly less negative with temperature increase and (?C _s /?T) _P and (?C ₄₄/?T) _P are almost constant. Combining the present elastic data with thermal expansion and specimen heat capacity data of CaO, we present the temperature dependence of thermodynamic parameters important in the studies of earth's interior.