Isochoric Heat Capacity Measurements for Light and Heavy Water Near the Critical Point

The isochoric heat capacity was measured for D₂O at a fixed density of 356.075 kg·m^–3 and for H₂O at 309.905 kg·m^–3. The measurements cover the range of temperatures from 623 to 661 K. The measurements were made with a high-temperature, high-pressure, adiabatic calorimeter with a nearly constant inner volume. The uncertainty of the temperature is 10 mK, while the uncertainty of the heat capacity is estimated to be 2 to 3%. Measurements were made in both the two-phase and the one-phase regions. The calorimeter instrumentation also enables measurements of PVT and the temperature derivative (P/T)_V along each measured isochore. A detailed discussion is presented on the experimental temperature behavior of C_V in the one- and two-phase regions, including the coexistence curve near the critical point. A quasi-static thermogram method was applied to determine values of temperature at saturation T_S() on measured isochores. The uncertainty of the phase-transition temperature measurements is about ±0.02 K. The measured C_V data for D₂O and H₂O are compared with values predicted from a recent developed parametric crossover equation of state and IAPWS-95 formulation.     

Isochoric Heat Capacity of CO2 + n-Decane Mixtures in the Critical Region

The isochoric heat capacity of two binary (CO₂+n-decane) mixtures (0.095 and 0.178 mole fraction of n-decane) have been measured with a high- temperature, high-pressure, nearly constant volume adiabatic calorimeter. Measurements were made at nineteen near-critical liquid and vapor densities between 87 and 658 kg·m⁻³ for the composition of 0.095 mole fraction n-decane and at nine densities between 83 and 458 kg·m⁻³ for the composition of 0.178 mole fraction n-decane. The range of temperatures was 295 to 568 K. These temperature and density ranges include near- and supercritical regions. The measurements were performed in both one- and two-phase regions including the vapor + liquid coexistence curve. The uncertainty of the heat- capacity measurements is estimated to be 2% (coverage factor k=2). The uncertainty in temperature is 15 mK, and that for density measurements is 0.06%. The liquid and vapor one- and two-phase isochoric heat capacities, temperatures (T _S), and densities (ρ_S) at saturation were measured by using the well-established method of quasi-static thermograms for each filling density. The critical temperatures (T _C), the critical densities (ρ_C), and the critical pressure (P _C) for the CO₂+n-decane mixtures were extracted from the isochoric heat-capacity measurements on the coexistence curve. The observed isochoric heat capacity along the critical isochore of the CO₂+n-decane mixture exhibits a renormalization of the critical behavior of C _{V X} typical for mixtures. The values of the characteristic parameters (K ₁, K ₂), temperatures (τ₁ ,τ₂), and the characteristic density differences were estimated for the CO₂+n-decane mixture by using the critical-curve data and the theory of critical phenomena in binary mixtures. The ranges of conditions were defined on the T-x plane for the critical isochore and the ρ-x plane for the critical isotherm, for which we observed renormalization of the critical behavior for the isochoric heat capacity.     

Isochoric Heat Capacity Measurements for Heavy Water Near the Critical Point

Isochoric heat capacity measurements of D₂O are presented as a function of temperature at fixed densities of 319.60, 398.90, 431.09, and 506.95 kg·m^–3. The measurements cover a range of temperatures from 551 to 671 K and pressures up to 32 MPa. The coverage includes one- and two-phase states and the coexistence curve near the critical point of D₂O. A high-temperature, high-pressure, adiabatic, and nearly constant-volume calorimeter was used for the measurements. Uncertainties of the heat capacity measurements are estimated to be 2 to 3%. Temperatures at saturation T _S () were measured isochorically using a quasi-static thermogram method. The uncertainty of the phase transition temperature measurements is about ±0.02 K. The measured C _V data for D₂O were compared with values predicted from a parametric crossover equation of state and six-term Landau expansion crossover model. The critical behavior of second temperature derivatives of the vapor pressure and chemical potential were studied using measured two-phase isochoric heat capacities. From measured isochoric heat capacities and saturated densities for heavy water, the values of asymptotic critical amplitudes were estimated. It is shown that the critical parameters (critical temperature and critical density) adopted by IAPWS are consistent with the T _S – _S measurements for D₂O near the critical point.     

Isochoric Heat-Capacity Measurements for Pure Methanol in the Near-Critical and Supercritical Regions

Isochoric heat-capacity measurements for pure methanol are presented as a function of temperature at fixed densities between 136 and 750 kg·m⁻³. The measurements cover a range of temperatures from 300 to 556 K. The coverage includes the one- and two-phase regions, the coexistence curve, the near-critical, and the supercritical regions. A high-temperature, high-pressure, adiabatic, and nearly constant-volume calorimeter was used for the measurements. Uncertainties of the heat-capacity measurements are estimated to be 2–3% depending on the experimental density and temperature. Temperatures at saturation, T _S(ρ), for each measured density (isochore) were measured using a quasi-static thermogram technique. The uncertainty of the phase-transition temperature measurements is 0.02 K. The critical temperature and the critical density for pure methanol were extracted from the saturated data (T _S,ρ_S) near the critical point. For one near-critical isochore (398.92 kg·m⁻³), the measurements were performed in both cooling and heating regimes to estimate the effect of thermal decomposition (chemical reaction) on the heat capacity and phase-transition properties of methanol. The measured values of C _V and saturated densities (T _S,ρ_S) for methanol were compared with values calculated from various multiparametric equations of state (EOS) (IUPAC, Bender-type, polynomial-type, and nonanalytical-type), scaling-type (crossover) EOS, and various correlations. The measured C _V data have been analyzed and interpreted in terms of extended scaling equations for the selected thermodynamic paths (critical isochore and coexistence curve) to accurately calculate the values of the asymptotical critical amplitudes ( and B ₀).     

Equation of State and Thermodynamic Properties of Pure D2O and D2O + H2O Mixtures in and Beyond the Critical Region

A parametric crossover model is adapted to represent the thermodynamic properties of pure D₂O in the extended critical region. The crossover equation of state for D₂O incorporates scaling laws asymptotically close to the critical point and is transformed into a regular classical expansion far from the critical point. An isomorphic generalization of the law of corresponding states is applied to the prediction of thermodynamic properties and the phase behavior of D₂O + H₂O mixtures over a wide region around the locus of vapor-liquid critical points. A comparison is made with experimental data for pure D₂O and for the D₂O + H₂O mixture. The equation of state yields a good representation of thermodynamic property data in the range of temperatures 0.8T _c(x)T1.5T _c(x) and densities 0.35_c(x)1.65_c(x).     

PVTx Measurements for a H2O + Methanol Mixture in the Subcritical and Supercritical Regions

PVTx relationships for a H₂O + CH₃OH mixture (0.36 mole fraction of methanol) were measured in a range of temperatures from 373 to 673 K and pressures between 0.042 and 90.9 MPa. The density ranged from 37.76 to 559.03 kg · m^–3. Measurements were made with a constant-volume piezometer surrounded by a precision thermostat. The temperature inside the thermostat was maintained uniform within 5 mK. The volume of the piezometer (32.68 ± 0.01 cm³) was previously calibrated from well-established PVT values of pure water (IAPWS), and was corrected for both temperature and pressure expansions. Uncertainties of the density, temperature, and pressure measurements are estimated to be 0.16%, 30 mK, and 0.05%, respectively. The uncertainty in composition is 0.001 mole fraction. The method of isochoric and isothermal break points was used to extract the phase transition temperatures, pressures, and densities for each measured isochore and isotherm. The values of the critical temperature, pressure, and density of the mixture were also determined from PVTx measurements in the critical region.     

Experimental Study of the Critical Behavior of the Isochoric Heat Capacity of Aqueous Ammonia Mixture

The isochoric heat capacity of a NH₃ + H₂O (0.2607 mole fraction of ammonia) mixture has been measured in the near- and supercritical regions. Measurements were made in the single- and two-phase regions including the coexistence curve using a high-temperature, high-pressure, nearly constant-volume adiabatic calorimeter. Measurements were made along 38 liquid and vapor isochores in the range from 120.03 kg · m⁻³ to 671.23 kg · m⁻³ and at temperatures from 478 K to 634 K and at pressures up to 28 MPa. Temperatures at the liquid–gas phase transition curve, T _S(ρ), for each measured density (isochore) and the critical parameters (T _C and ρ _C) for the 0.2607 NH₃ + 0.7393  H₂O mixture were obtained using the quasi-static thermograms technique. The expanded uncertainty of the heat-capacity measurements at the 95 % confidence level with a coverage factor of k = 2 is estimated to be 2 % to 3 % in the near-critical and supercritical regions, 1.0 % to 1.5 % in the liquid phase, and 3 % to 4 % in the vapor phase. Uncertainties of the density, temperature, and concentration measurements are estimated to be 0.06 %, 15mK, and 5×10⁻⁵ mole fraction, respectively. An unusual behavior of the isochoric heat capacity of the mixture was found near the maxcondetherm point (in the retrograde region). The value of the Krichevskii parameter was calculated using the critical properties data for the mixture and vapor-pressure data for the pure solvent (H₂O). The derived value of the Krichevskii parameter was used to analyze the critical behavior of the strong (C _P , K _T ) and weakly (C _V ) singular properties in terms of the principle of isomorphism of critical phenomena in binary mixtures. The values of the characteristic parameters (K ₁, K ₂), temperatures (τ ₁, τ ₂), and the characteristic density differences (Δρ ₁, Δρ ₂) were calculated for the NH₃ + H₂O mixture by using the critical-curve data.     

Thermodynamic Properties of Methanol in the Critical and Supercritical Regions

The isochoric heat capacity of pure methanol in the temperature range from 482 to 533 K, at near-critical densities between 274.87 and 331.59 kg· m⁻³, has been measured by using a high-temperature and high-pressure nearly constant volume adiabatic calorimeter. The measurements were performed in the single- and two-phase regions including along the coexistence curve. Uncertainties of the isochoric heat capacity measurements are estimated to be within 2%. The single- and two-phase isochoric heat capacities, temperatures, and densities at saturation were extracted from experimental data for each measured isochore. The critical temperature (T_c = 512.78±0.02K) and the critical density (ρ_c = 277.49±2 kg · m⁻³) for pure methanol were derived from the isochoric heat-capacity measurements by using the well-established method of quasi-static thermograms. The results of the C_VVT measurements together with recent new experimental PVT data for pure methanol were used to develop a thermodynamically self-consistent Helmholtz free-energy parametric crossover model, CREOS97-04. The accuracy of the crossover model was confirmed by a comprehensive comparison with available experimental data for pure methanol and values calculated with various multiparameter equations of state and correlations. In the critical and supercritical regions at 0.98T_c≤ T ≤ 1.5T_c and in the density range 0.35ρ_c ≤ ρ leq 1.65 ρ_c, CREOS97-04 represents all available experimental thermodynamic data for pure methanol to within their experimental uncertainties.     

Isochoric Heat Capacity of Heavy Water at Subcritical and Supercritical Conditions

The heat capacity of heavy water was measured in the temperature range from 294 to 746 K and at densities between 52 and 1105 kg·m^–3 using a high-temperature, high-pressure adiabatic calorimeter. The measurements were performed at 14 liquid and 9 vapor densities between 52 and 1105 kg·m^–3. Uncertainties of the measurements are estimated to be within 3% for vapor isochores and 1.5% for the liquid isochores. In the region of the immediate vicinity of the critical point (0.97T/T _c1.03 and 0.75/_c1.25), the uncertainty is 4.5%. The original C _V data were corrected and converted to the new ITS-90 temperature scale. A parametric crossover equation of state was used to represent the isochoric heat capacity measurements of heavy water in the extended critical region, 0.8T/T _c1.5 and 0.35/_c1.65. The liquid and vapor one- and two-phase isochoric heat capacities, temperatures, and saturation densities were extracted from experimental data for each measured isochore. Most of the experimental data are compared with the Hill equation of state, and the overall statistics of deviations between experimental data and the equation of state are given.     

Saturation vapor pressure and critical constants of H2O,D2O,T2O,and their isotopic mixtures

Reliable data on the vapor pressure and critical constants of H₂O isotopes and their isotopic mixtures are required for the generation of thermophysical properties data over a wide range of temperatures and pressures. In this study, vapor pressure equations for D₂O and T₂O have been developed based on the latest experimental and theoretical information. Considering the similarity among H₂O isotopes, the functional form of the Saul and Wagner equation, fully proven for H₂O, has been employed. The present equation for D₂O shows a lower trend by up to 0.09% than the widely used Hill and MacMillan equation at temperatures below 150°C. For the vapor pressure of the isotopic mixtures, the available experimental data have been examined for the validity of Raoult's law. Then it has been shown that the critical temperature and the critical pressure of the isotopic mixture can also be predicted as simple mole-fraction average values.     

Two-Phase Isochoric Heat Capacity Measurements for Nitrogen Tetroxide in the Critical Region and Yang–Yang Relation

The two-phase isochoric heat capacity of nitrogen tetroxide was measured in the temperature range from 261.74 K to the critical temperature (431.072 K) at densities between 201.21 and 1426.5 kg·m^–3 using a high-temperature and high-pressure adiabatic calorimeter. The measurements were performed in the two-phase region for 26 isochores (15 liquid and 11 vapor densities) including the coexistence curve and critical region. Uncertainties of the measurements are estimated to be 2%. The original temperatures and C _V data were converted to the ITS-90. The liquid and vapor two-phase isochoric heat capacities, temperatures, and densities at saturation were extracted from experimental data for each measured isochore. From measured (T _S, _S, C _V2, C _V2) data, the values of second temperature derivatives of vapor-pressure d ² P _S/dT ² and chemical potential d ² /dT ² were derived using the Yang–Yang relation. The results were compared with values calculated from other vapor-pressure equations. The values of saturated densities and critical parameters derived in calorimetric experiments were compared with literature data. The unusual temperature behavior of d ² P _S/dT ² and d ² /dT ² was found at low temperatures around 351 K and near the critical point.     

On the Interpretation of Near-Critical Gas–Liquid Heat Capacities by L. V. Woodcock,Int. J. Thermophys. (2017) 38, 139: Comments

It is shown that: (1) the expressions for the isochoric (C_V) and isobaric (C_P) heat capacities of liquid and gas, coexisting in phase equilibrium, the heat capacities at saturation of liquid and gas (C_σ) and the heat capacity C_λ used in the article “On the Interpretation of Near-Critical Gas–Liquid Heat Capacities, L. V. Woodcock, Int. J. Thermophys. (2017) 38, 139” are incorrect; (2) the conclusions of the article based on the comparison of the incorrect C_V, C_P and C_λ with experimental data are also incorrect; (3) the lever rule cannot be used to define C_V and C_P in the two-phase coexistence region; (4) a correct expression for the isochoric heat capacity describes well the experimental data; (5) there is no misinterpretation of near-critical gas–liquid heat capacity measurements in the two-phase coexistence region; (6) there are no proofs in the article that: (a) the divergence of C_V is apparent; (b) it has not been established experimentally that the thermodynamic properties of fluids satisfy scaling laws with universal critical exponents asymptotically close to a single critical point of the vapor–liquid phase transition; and (c) there is no singular critical point on Gibbs density surface. We obtained the relations connecting the isochoric heat capacity in the two-phase region with thermodynamic properties at saturation of homogeneous liquid and gas which can be used to verify the equation of state.     

Heat capacity studies of ThP and UP0.5As0.5 solid solution. Reanalyzing UP data

The specific heats of ThP and the solid solution UP_0.5As_0.5 have been measured in the temperature range 5–300 K. While ThP has a regularC _p (T) behavior, the mixed compound exhibits several low-temperature anomalies. An analysis of the experimental data for UP_0.5As_0.5 and reanalysis of previously published heat capacity results for UP have been performed. The temperature dependence of the magnetic entropy, which at 300 K reaches a value close toR ln 4 for both species, confirms the U³⁺ state for uranium atoms in these compounds. The value of the electronic heat capacity coefficient _p in the paramagnetic state has also been extracted from the experimental data. It is far smaller than the respective low-temperature value (0).     

The melting curve of tetrahydrofuran hydrate in D2O

Melting points for the tetrahydrofuran/D₂O hydrate in equilibrium with the airsaturated liquid at atmospheric pressure are reported. The melting points were measured by monitoring the absorbance of the solution. Overall, the meltingpoint phase boundary curve is about 2.5 K greater than the corresponding curve for the H₂O hydrate, with a congruent melting temperature of 281±0.5 K at a D₂O mole fraction of 0.936. The phase boundary is predicted to within 5% if the assumption is made that the THF occupancy in the D₂O and H₂O hydrates is the same. We measure an occupancy of 99.9%. The chemical potential of the empty lattice in D₂O is estimated to be 5% greater than in H₂O.     

Liquid mutual diffusivities of the H2O/D2O system

The liquid-phase mutual diffusivities of the water (H₂O) and deuterium oxide (D₂O) system at 298.2 K were measured using an instrument based on the Taylor dispersion technique. The instrument has been designed to match, as closely as possible, the mathematical model of ideal Taylor dispersion, minimizing all the departures from the ideal model. The diffusivities were measured over the entire concentration range and the results follow a linear dependence on molar fraction given by 10⁹ D ₁₂ = , where D ₁₂ is in m²·s^–1. Comparison with highly accurate data obtained by a Rayleigh interferometer seems to indicate that the accuracy of the present instrument is 1%. The hard-sphere model was applied to the estimation of the mutual diffusivities of this system and good agreement was found with experiment, deviations being ±3.5%.Paper presented at the Ninth Symposium on Thermophysical Properties, June 24–27, 1985, Boulder, Colorado, U.S.A.     

Crystallization and properties of CaO-P2O2-B2O3 glasses

Glasses were prepared with compositions (50–0.5 x) CaO.(50–0.5 x) P₂O₅ · x B₂O₃ with B₂O₃ contents (x) from 0 to 45 mol%. The glass transformation temperature (T _g), dilatational softening temperature (T _D) and Vickers hardness (H _V) initially increased with x, but showed maxima at about x=20 for T _g and T _D and at about x=35 for H _V. The thermal expansion coefficient decreased with x, levelling off at about 35 mol% B₂O₃. The maximum tendency to crystallize occurred at around 25 mol% B₂O₃. Volume nucleation (and hence glass-ceramic formation) and surface nucleation were obtained for x between 15 and 25 mol%. The first phase to appear was BPO₄, which was probably homogeneously nucleated. Subsequently the 4CaO · P₂O₅ phase was heterogeneously nucleated on the BPO₄. For 10 x 35 only surface nucleation was observed. The kinetics of nucleation were investigated in the 20 mol% B₂O₃ glass. The changes in properties and crystallisation behaviour with B₂O₃ content were related to short-range structural information. Infrared spectra and literature data indicated a threedimensional network of B-O-B and B-O-P linkages in the glasses.     

Influence of ions on the critical behavior of a binary mixture near the consolute point

The lower critical point of stratification of a 3-methylpyridine (MP)+heavy water (D₂O) mixture in the presence of Na⁺ and CI^– ions has been studied by the Toepler shadow method. Addition of 0.3% ions lowered the critical temperature and reduced the equilibration time and the gradient of the refractive index (compressibillity). The analysis of the form of the near-critical isotherm demonstrated the ionic mixture to be described by the index = 3.05 ± 0.15. which corresponds to the classical mean-field theory. The results obtained provide evidence that even small admixtures of charged particles result in a substantial suppression of fluctuations near the critical point by the long-range Coulomb interaction.Paper presented at the Twelfth Symposium on Thermophysical Properties, June 19–24, 1994, Boulder, Colorado, U.S.A.     

Decay rate of critical fluctuations in steam and in dilute steam-NaCl mixtures

The decay rate of critical fluctuations in steam and in a steam-NaCl mixture has been investigated experimentally with the aid of photon correlation spectroscopy. For pure steam, the measurements have been performed along seven isochores [(¦– _c¦)/_c<0.09] as a function of the temperatureT for (T–T_t)<1 K. The results have been compared with the values predicted by the renormalization-group theory written as a modification of the classical mode coupling theory. The agreement between experiment and theory is satisfactory along the critical isochore, but larger deviations are noted for _c when approching the transition temperatureT _t. The decay rate of a 0.1% (molar) dilute mixture of NaCl in H₂O has been measured along some near-critical isochores as a function of temperature. Its behavior, which is very different from that observed for pure steam, is dicussed.Paper presented at the Tenth Symposium on Thermophysical Properties, June 20–23, 1988, Gaithersburg, Maryland, U.S.A.     

Experimental Study of the Isochoric Heat Capacity of Diethyl Ether (DEE) in the Critical and Supercritical Regions

Two- and one-phase liquid and vapor isochoric heat capacities (C _{V ρ} T relationship) of diethyl ether (DEE) in the critical and supercritical regions have been measured with a high-temperature and high-pressure nearly constant-volume adiabatic calorimeter. The measurements were carried out in the temperature range from 347 K to 575 K for 12 liquid and 5 vapor densities from 212.6 kg·m⁻³ to 534.6 kg·m⁻³. The expanded uncertainties (coverage factor k =  2, two-standard deviation estimate) for values of the heat capacity were 2% to 3% in the near-critical region, 1.0% to 1.5% for the liquid isochores, and 3% to 4% for the vapor isochores. The uncertainties of density (ρ) and temperature (T) measurements were 0.02% and 15 mK, respectively. The values of the internal energy, U(T, V), and second temperature derivative of pressure, (∂² P/∂T ²) _ρ , were derived using the measured C _V data near the critical point. The critical anomaly of the measured C _V and derived values of U(T, V) and (∂² P/∂T ²) _ρ in the critical and supercritical regions were interpreted in terms of the scaling theory of critical phenomena. The asymptotic critical amplitudes (A₀⁺ and A₀^- ){({A_0^+} {\rm and} {A_0^- )}} of the scaling power laws along the critical isochore for one- and two-phase C _V were calculated from the measured values of C _V . Experimentally derived values of the critical amplitude ratio for C_V (A₀⁺ /A₀^- = 0.521){C_{V} \left({A_0^+ /A_0^- = 0.521}\right)} are in good agreement with the values predicted by scaling theory. The measured C _V data for DEE were analyzed to study the behavior of loci of isothermal and isochoric C _V maxima and minima in the critical and supercritical regions.