Excess Heat Capacity Of Mixing Research Articles

Excess heat capacity and entropy of mixing along the hydroxyapatite-chlorapatite and hydroxyapatite-fluorapatite binaries

The heat capacity, Cp, of synthetic hydroxyapatite [Ca5(PO4)3OH–OH-Ap], as well as of ten compositions along the OH-Ap-chlorapatite (Cl-Ap) join and 12 compositions along the OH-Ap-fluorapatite (F-Ap) join have been measured using relaxation calorimetry (heat capacity option of the Physical Properties Measurement System—PPMS) and differential scanning calorimetry (DSC) in the temperature range of 5–764 K. Apatites along the Cl-OH and F-OH joins were synthesized at 1100 °C and 300 MPa in an internally heated gas pressure vessel via an exchange process between synthetic fluorapatite or chlorapatite crystals (200–500 μm size) and a series of Ca(OH)2-H2O solutions with specific compositions and amounts relative to the starting apatite. The standard third-law entropy of OH-Ap, derived from the low-temperature heat capacity measurements, is S° = 386.3 ± 2.5 J mol−1 K−1, which is ~ 1% lower than that resulting from low-temperature adiabatic calorimetry data on OH-Ap from the 1950’s. The heat capacity of OH-Ap above 298.15 K shows a hump-shaped anomaly centred around 442 K. Based on published structural and calorimetric work, this feature is interpreted to result from a monoclinic to hexagonal phase transition. Super ambient Cp up to this transition can be represented by the polynomial: C_{p}^{{text{OH - Ap}}} {}_{{298K - 442K}}left( {{text{J mol}}^{ - 1} {text{K}}^{- 1}} right) = {1013.7-13735.5T^{{ - 0.5}}} + 2.616718,10^{7} T^{{ - 2}} - 3.551381,10^{9} T^{{ - 3}} .. The DSC data above this transition were combined with heat capacities computed using density functional theory and can be given by the Cp polynomial: C_{p}^{{text{OH - Ap}}} {}_{{ >,442K}}left( {{text{J mol}}^{ - 1} {text{K}}^{- 1}} right) = {877.2-11393.7 T^{ - 0.5}} + {5.452030,10^{7}} ,T^{- {2}} - {1.394125,10^{10}} ,T^{- {3}}. Positive excess heat capacities of mixing, ∆Cpex, in the order of 1–2 J mol−1 K−1, occur in both solid solutions at around 70 K. They are significant at these conditions exceeding the 2σ-uncertainty of the data. This positive ∆Cpex is compensated by a negative ∆Cpex of the same order at around 250 K in both binaries. At higher temperatures (up to 1200 K), ∆Cpex is zero within error for all solid solution members. As a consequence, the calorimetric entropies, Scal, show no deviation from ideal mixing behaviour within a 2σ-uncertainty for both joins. Excess entropies of mixing, ∆Sex, are thus zero for the OH-Ap–F-Ap, as well as for the OH-Ap–Cl-Ap join. The Cp–T behaviour of the OH-Ap endmember is discussed in relation to that of the F- and Cl-endmembers.

First-principles investigation of the lattice vibrations in the alkali feldspar solid solution.

The heat capacities of Al, Si ordered alkali feldspars of different Na, K compositions were calculated using the density functional theory. The effect of the Na, K distribution, if random, ordered or clustered, on the resulting heat capacity was investigated on different cells with Ab50Or50 composition. For all compositions and distributions studied, the excess heat capacity of mixing is positive at low temperatures with a maximum at ~60 K. This produces an increasing excess vibrational entropy that reaches a constant value above ~200 K. The amount of the excess heat capacity of Ab50Or50, however, depends on the Na, K distribution. Best agreement with measured excess heat capacities is achieved, if the distribution of Na and K is either ordered or clustered. The positive excess heat capacities can be attributed to a strong softening of the acoustic and the lowest optical modes related to a strong increase of Na–O bond lengths in samples with intermediate compositions. The softening of the lowest optical mode is, however, not reflected by thoroughly measured literature IR data. Comparing calculated and measured IR spectra suggests that the resolution of the measured spectra was insufficient for detecting the lowest IR-active modes.Electronic supplementary materialThe online version of this article (doi:10.1007/s00269-014-0715-8) contains supplementary material, which is available to authorized users.

On the Tendency of Solutions to Tend Toward Ideal Solutions at High Temperatures

The rule of Lupis and Elliott (LE rule) proposed for the first time in 1966 is reformulated in this article as, “Real solid, liquid and gaseous solutions (and pure gases) gradually approach the state of an ideal solution (perfect gas) as temperature increases at any fixed pressure and composition.” This rule is rationalized through the heat expansion of phases and loss of any interaction with increased separation between the atoms. It is shown that the rule is valid only if the standard state is selected properly, i.e., if mixing does not involve any hidden phase changes, such as melting. It is shown that the necessary and sufficient practical conditions to obey the LE rule is the equality of signs of the heat of mixing and excess entropy of mixing and the nonequality of signs of heat of mixing and excess heat capacity of mixing of the same solution. It is shown that these two conditions are fulfilled for most of the experimentally measured high-temperature solutions. The LE rule is compared with the existing laws of thermodynamics. It is shown that the LE rule can be considered as a potential fourth law of materials thermodynamics.

On the nature of the excess heat capacity of mixing

The excess vibrational entropy (ΔSvibex) of several silicate solid solutions are found to be linearly correlated with the differences in end-member volumes (ΔVi) and end-member bulk moduli (Δκi). If a substitution produces both, larger and elastically stiffer polyhedra, then the substituted ion will find itself in a strong enlarged structure. The frequency of its vibration is decreased because of the increase in bond lengths. Lowering of frequencies produces larger heat capacities, which give rise to positive excess vibrational entropies. If a substitution produces larger but elastically softer polyhedra, then increase and decrease of mean bond lengths may be similar in magnitude and their effect on the vibrational entropy tends to be compensated. The empirical relationship between ΔSvibex, ΔVi and Δκi, as described by ΔSvibex = (ΔVi + mΔκi)f, was calibrated on six silicate solid solutions (analbite–sanidine, pyrope–grossular, forsterite–fayalite, analbite–anorthite, anorthite–sanidine, CaTs–diopside) yielding m = 0.0246 and f = 2.926. It allows the prediction of ΔSvibex behaviour of a solid solution based on its volume and bulk moduli end-member data.

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

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

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

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

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

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

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

Low- and high-temperature heat capacities for a series of synthetic high structural state plagioclase crystals (Ab-An) were measured using both a relaxation calorimeter and a differential scanning calorimeter. The measurements were performed at temperatures between 5 and 800 K on milligram-sized polycrystalline samples that had been characterized in a previous study. The data show positive excess heat capacities of mixing at temperatures below 300 K with a maximum value of ~2 J/(mol·K). Below ~70 K, the excess heat capacities exceed two standard deviations and are thus significant. Above 300 K, the measurements indicate negative excess heat capacities with a maximum of ca. −1.5 J/(mol·K) at about 400 K, and do not exceed two standard deviations. The excess vibrational entropies of mixing are positive with an asymmetric variation. At T = 298.15 K, the largest deviation from ideal behavior occurs at Ab 20 An 80 amounting to Δ S vib ex = 2.8 ± 2.4 J/(mol·K). An asymmetric Margules mixing model was found to adequately describe the vibrational entropy-composition behavior, yielding W AbAn Svib = 16.4 J/(mol·K) and = W AbAn Svib 4.7 J/(mol·K).

Entropies of mixing and subsolidus phase relations of forsterite-fayalite (Mg2SiO4-Fe2SiO4) solid solution

The heat capacities of a series of synthetic forsterite (Fo).fayalite (Fa), Mg 2 SiO 4 .Fe 2 SiO 4 , olivines have been measured between 5 and 300 K on milligram-sized samples with the Physical Properties Measurement System (Quantum Design). The heat capacities for fayalite and fayalite-rich olivine are marked by a sharp lambda-type anomaly defining a transition from the paramagnetic to an antiferromagentic state, which in the case of fayalite occurs at T N = 64.5 K. In forsterite-rich compositions a feature in the C P data around 25 K is observable and it could possibly be linked to a magnetic transition. Additionally, all Fe-bearing olivines show a Schottky-type anomaly. Excess heat capacities of mixing, ΔC xs P , for the various Fe-Mg olivine solid-solution compositions were calculated applying the equation ΔC xs P = C ss P -[(1 - XFa) C P o + X Fa C P Fa ] using fitted C P polynomials for each composition. The calorimetric entropies at 298.15 K, Scal, were determined by solving the C P integral . If a symmetric Margules mixing model ΔS xs = Ws·X Fa (1 -XFa) is taken to describe the entropy of mixing behavior for the Fo-Fa binary, it yields an interaction parameter of W s = -1.6 ± 1.7 J/(mol·K) on a onecation basis. The calorimetric data thus indicate ideal entropy of mixing behavior. Adopting, however, a value of W Ol S,Mg-Fe = -1.6 J/(mol·K) one can calculate a value for the excess Gibbs free energy of mixing of W Ol G,Mg-Fe = 6.9 kJ/mol at 1000 K using the most recent solution calorimetric study of Kojitani and Akaogi (1994) on Fo-Fa olivine with W Ol H,Mg-Fe = 5.3 kJ/mol. This WOl G,Mg-Fe value should be considered a maximum upper limit for thermodynamic nonideality. Using solely calorimetric data, the T-X phase diagram for the Fo-Fa binary is calculated at 1 bar and 50 kbar and compared to that obtained from a model-dependent thermodynamic analysis. The results suggest that exsolution in Fe-Mg olivine should only be possible in low-temperature environments depending on kinetic behavior.

Thermodynamics of alloys and phase equilibria in the copper-iron system

Together with previous values at 1873 K, new measurements of the enthalpies of formation of liquid Cu-Fe alloys, carried out by high-temperature calorimetry at 1673 K, have revealed a temperature dependence of ΔH, corresponding to a negative excess heat capacity of mixing. Our calorimetric results were combined with critically selected values of the enthalpies of alloying and activities of components in liquid and solid solutions, as well as parameters of stable and metastable phase transitions, to perform a thermodynamic evaluation of the system. The latter was carried out in the spirit of the CALPHAD approach, using Thermo-Calc software. The thermodynamic model generates a self-consistent description of all thermodynamic properties and phase equilibria, including metastable solidification, in close agreement with reliable experimental data. In particular, a very satisfactory representation of stable (L+δ)/γ, (L + γ)/e, γ/(e + α) and metastable (L1/L2), (L1 + L2)/e phase boundaries has been achieved. The occurrence of a monotectic reaction L2 ↔ L1 + δ is suggested in the region of the metastable precipitation of δ-phase. A comprehensive thermodynamic model of the Cu-Fe system is used to explain the widening of the concentration limits of formation of supersaturated bcc and fcc solutions prepared by highly nonequilibrium methods of synthesis.

Thermal properties of melt-blended poly(ether ether ketone) and poly(ether imide)

The thermal properties of blends of poly(ether ether ketone) (PEEK) and poly(ether imide) (PEI) prepared by screw extrusion were investigated by differential scanning calorimetry. From the thermal analysis of amorphous PEEK–PEI blends which were obtained by quenching in liquid nitrogen, a single glass transition temperature (Tg) and negative excess heat capacities of mixing were observed with the blend composition. These results indicate that there is a favorable interaction between the PEEK and PEI in the blends and that there is miscibility between the two components. From the Lu and Weiss equation and a modified equation from this work, the polymer–polymer interaction parameter (χ12) of the amorphous PEEK–PEI blends was calculated and found to range from −0.058 to −0.196 for the extruded blends with the compositions. The χ12 values calculated from this work appear to be lower than the χ12 values calculated from the Lu and Weiss equation. The χ12 values calculated from the Tg method both ways decreased with increase of the PEI weight fraction. © 1999 John Wiley & Sons, Inc. J Appl Polym Sci 72: 733–739, 1999

Heat capacity measurements of synthetic pyrope-grossular garnets between 320 and 1000 K by differential scanning calorimetry

The heat capacity of synthetic periclase (MgO), pyrope (M93Al2Si3O12), grossular (Ca3Al2Si3O12), and four pyrope-grossular solid solutions (PY89.8Gr10.2, PY73.6Gr26.4, PY48.8Gr51.2, PY24.0Gr76.0) has been measured between 320 and 1000 K by differential scanning calorimetry. Measurements were made in both interval-scanning and step-scanning modes; with the latter method, Cp was determined to a precision of better than ± 1%. The results, together with previous calorimetric measurements on synthetic grossular and pyrope, now permit a complete description of the heat capacities for both phases between 10–1200 K and 10–1350 K, respectively. The following best-fit equations for the temperature range from 298 to 1200 K were obtained for pyrope and grossular, respectively (T in Kelvin and Cp in J/ mol∗K): Cp(pyrope) = 542.01 − 1387.07·T−0.5 − 20.492×106·T−2+2500.67×106·T−3; Cp(grossular) = 607.81 − 3214.49·T−0.5 − 12.141×106·T−2+1269.92×106·T−3.For the solid solution garnets a model assuming ideal mixing (linear combination) of molar heat capacities of the endmember phases reproduces the experimental heat capacities within errors. This indicates that the vibrational entropies of mixing and enthalpies of mixing are temperature-independent between room temperature and 1000 K. It is expected that excess heat capacities of mixing are not present in most silicate solid solutions at temperatures between 320 and 1000 K.

Glass transitions in hydrogen‐bonded polymer complexes

AbstractMutual precipitates of poly (N, N‐dimethyl acrylamide) and poly (4‐hydroxystyrene) were collected from dioxane, methanol, or acetone. The glass transition (Tg) temperatures of the precipitates are higher than the weight‐average values. Clear films cast from dimethylformamide solutions have lower Tg values. Complexation also occurred between poly (ethyl oxazoline) and poly (4‐hydroxystyrene) in dioxane and between poly (vinyl pyrrolidone) and poly (4‐hydroxystyrene) in methanol. Again, the glass transition temperatures of the precipitates are higher than the values for the blend films. The ΔCp values associated with the glass transitions of the complexes are smaller than those of the blends having the same compositions. Negative excess heat capacities of mixing have been observed for several precipitates.

Calorimetric study of binary systems of tetraethyleneglycol octylether and polyethyleneglycol with water

Calorimetric measurements have been made of the differential enthalpies of solution as a function of composition of both components in the binary systems tetraethyleneglycol octylether (C8E4)-water and polyethyleneglycol 400 (PEG)-water, as a function of composition, at three different temperatures. Heat capacity changes for dissolution were calculated from the temperature variation of the solution enthalpies. Excess enthalpies and excess heat capacities of mixing were calculated from the differential enthalpies of solution. All measurements on C8E4 were made above the critical micelle concentration (c.m.c.) so the results relate to C8E4 in aggregated form. The thermochemical properties of the C8E4 and PEG systems with water are similar. The differential solution enthalpy of the organic solute in pure water is fairly exothermic and then increases smoothly with increasing solute content. Likewise the solution enthalpy of water in pure C8E4 or PEG is fairly exothermic, but increases steadily to become zero at a water content corresponding to more than five water molecules per ethyleneoxide group. The measurements on the C8E4 system at 40°C were made close to the demixing temperature. The results are compared with previously reported results on the 2-butoxyethanol (BE)-water system.

Analytical Expressions for the Heat Capacities of Alkali Silicate Liquids

The available thermochemical data for silicate liquids and glasses indicate that the customary assumption of heat‐capacity additivity is not valid, particularly for liquid silicates near 1000 K. Estimates are obtained for the heat capacities of KO0.3‐SiO2, NaO0.3‐SiO2, and CaO‐SiO2 liquids. The excess heat capacity of mixing of alkali silicates at 1000 K is: image in units of cal mol−1 K−1. Analytical expressions are given for Cep for KO0.5‐NaO0.5‐SiO2 and NaO0.3‐CaO‐SiO2 liquids at 900 to 1800 K.

The enthalpy of sodium silicate glasses and liquids

The enthalpy data of Hummel and Schwiete (1) for sodium silicate glasses at 298.15 K have been fitted by power series in the mole fraction. It is shown that four coefficients provide an excellent fit of the data (X 1 = .1818 to .6667) but five are required for a reasonable extrapolation to pure NaO .5(X 1 = 1). If estimates for the partial molal enthalpy of SiO 2 are included in the least squares, the equation also extrapolates well to pure SiO 2. Combining these values with the expressions for the excess heat capacity of mixing (2) and known solubility data (3,4) for the binary system, one obtains a consistent set of coefficients for the calculation of enthalpies and activities at any temperature and composition. The NaO .5-SiO 2 phase diagram calculated from these values is shown.

Enthalpy-concentration diagram for the system hydrogen peroxide-water at 50 mm Hg total pressure

An enthalpy-concentration diagram of the binary system hydrogen peroxide-water was constructed at 50 mm Hg total pressure. Lack of information on heat capacities of the solutions at different temperatures necessitated the author assuming that the excess heat capacity of mixing remains constant in the temperature range required in this diagram. Isobaric liquid—vapour equilibrium data were also calculated at 50 mm Hg from the isothermal data of Scatchard, Kavanagh and Ticknor [1]. The results of these calculations are presented in a Figure.