Neumann-Kopp Rule Research Articles

Thermodynamic evaluation of the BaO-ZrO2-YO1.5 system

Based upon the experimental data available in the literature and present measurements, thermodynamic reassessments were initially performed on the binary BaO-ZrO2, BaO-YO1.5 systems, and the resultant thermodynamic parameters were then merged in combination with that of the previous ZrO2-YO1.5 system to derive a self-consistent thermodynamic description for the ZrO2-BaO-YO1.5 ternary system with limited ternary thermodynamic parameters. The Gibbs energies of all the liquid and terminal solid solution phases were treated by the substitutional solution model, the BZ phase featured by second solid solution described by the compound energy formalism (CEF) model, and the stoichiometric compound BZY424 and Ba2YZrO6-d modeled following the Neumann-Kopp rule. The calculated results agree well with the isothermal sections at 1600°C and 1750°C (1873K and 1923K) from both of the previous and present measured phase diagrams. This demonstrates that the thermodynamic parameters derived in the present work could be applicable to compositional optimizations of novel refractories for melting titanium alloys on the basis of this ternary oxide system.

The field theory of specific heat

Finite temperature quantum field theory in the heat kernel method is used to study the heat capacity of condensed matter. The lattice heat is treated a la P. Debye as energy of the elastic (sound) waves. The dimensionless functional of free energy is re-derived with a cut-off parameter and used to obtain the specific heat of crystal lattices. The new dimensionless thermodynamical variable is formed as the Planck's inverse temperature divided by the lattice constant. The dimensionless constant, universal for a class of crystal lattices, which determines the low temperature region of molar specific heat, is introduced and tested with the data for diamond lattice crystals. The low temperature asymptotics of specific heat is found to be the fourth power in temperature instead of the cubic power law of the Debye theory. Experimental data for the carbon group elements (silicon, germanium) and other materials decisively confirm the quartic law. The true low temperature regime of specific heat is defined by the surface heat, therefore, it depends on geometrical characteristics of a body, while the absolute zero temperature limit is geometrically forbidden. The limit on a growth of specific heat at temperatures close to critical points, known as the Dulong-Petit law, appears from the lattice constant cut-off. Its value depends on the lattice type and it is the same for materials with the same crystal lattice. The Dulong-Petit values of compounds are equal to those of elements with the same crystal lattice type, if one mole of solid state matter were taken as the Avogadro number of the lattice atoms. This means the Neumann-Kopp rule is valid only in some cases.

Thermodynamic description of the AgCl–CoCl2–InCl3–NaCl system

Critical thermodynamic evaluations and optimizations of the AgCl–CoCl2, AgCl–InCl3, AgCl–NaCl, InCl3–NaCl and CoCl2–InCl3 systems were carried out in the present work within the framework of the CALPHAD approach. The molten salt phase was described by the Modified Quasi-chemical Model, while the Compound Energy Formalism with various lattice ratios was used to treat the terminal solid solutions. The two binary double salts (Na3InCl6 and Ag3InCl6) were treated as the stoichiometric compounds of which Gibbs energies were modelled following the Neumann-Kopp rule. All the model parameters were optimized in terms of the required phase equilibria and thermochemical data arising from experimental measurements and theoretical predictions (First-principles method and empirical equation). A set of self-consistent thermodynamic database for the AgCl–CoCl2–InCl3–NaCl quaternary system was finally derived by merging the Gibbs energies for relevant phases in all binary systems (model parameters of CoCl2–NaCl were adopted from the literature) based upon the Kohler–Toop interpolation technique in order to facilely calculate various phase equilibria and thermodynamic properties of the sub-ternary and quaternary systems. The calculated results involving multicomponent systems will support beneficial instructions in the related industrial processes.

Thermodynamic assessment of B2O3–MgO binary system

The B2O3–MgO system has been thermodynamically assessed by CALculation of PHAse Diagram (CALPHAD) technique using Thermo-Calc software. An ionic two-sublattice (I2SL) model, (Mg2+)p(O2−, BO33-, B4O72-, B3O4.5)Q, is proposed to describe the liquid phase in the system, whereas the Redlich–Kister expression is used to formulate its excess Gibbs energy. All the solid intermediate compounds have been treated as stoichiometric compounds and their Gibbs energy is modeled following Neumann–Kopp's rule. A set of self-consistent thermodynamic parameters for the B2O3–MgO system are then obtained and the calculated phase diagram and thermodynamic properties of the B2O3–MgO system agree well with the available phase diagram and experimental data in the literature.

Specific Heat Capacity Measurements of Intermetallic Phases in the Ternary Al-Ti-Cr System

Specific heat capacities of the γ-TiAl and τ phases in the Al-Ti-Cr system were determined in the temperature range between 298 and 773/873 K based on one binary (51.2Al-48.8Ti) and three ternary alloys (53Al-44Ti-3Cr, 64.6Al-30.2Ti-5.2Cr, 65.3Al-26.7Ti-8Cr). The specific heat capacity measurements in this system were performed for the first time. The measurements were carried out using heat-flux DSC devices employing the classical three step method. The measured heat capacities were lower than the approximation using the Neumann-Kopp rule. These differences in the heat capacity data arise probably due to the variation of the chemical bonding state between the intermetallic phases and the pure elements which are not taken into account by Neumann-Kopp’s rule. In order to check the reliability of the obtained results, the enthalpy increment was measured by using drop calorimetry. The heat increments determined using the drop calorimetry was in an excellent agreement with the enthalpy increments calculated using the analytic description of the heat capacities from DSC.

Enthalpies of formation of Cd–Pr intermetallic compounds and thermodynamic assessment of the Cd–Pr system

In the present study standard enthalpies of formation were measured by reaction and solution calorimetry at stoichiometric compositions of Cd2Pr, Cd3Pr, Cd58Pr13 and Cd6Pr. The corresponding values were determined to be −46.0, −38.8, −35.2 and −24.7kJ/mol(at), respectively. These data together with thermodynamic data and phase diagram information from literature served as input data for a CALPHAD-type optimization of the Cd–Pr phase diagram. The complete composition range could be described precisely with the present models, both with respect to phase equilibria as well as to thermodynamic input data. The thermodynamic parameters of all intermetallic compounds were modelled following Neumann–Kopp rule. Temperature dependent contributions to the individual Gibbs energies were used for all compounds. Extended solid solubilities are well described for the low- and high-temperature modifications of Pr and also for the intermetallic compound CdPr. A quite good agreement with all viable data available from literature was found and is presented.

Structure and heat capacity of the NaCeF4 compound

This work continued our general research program on obtaining metallic cerium by electrodeposition from NaCeF4 dissolved in different molten fluorides. The structure of NaCeF4 (cubic or hexagonal depending on the way of preparation) was established by DTA analysis, IR spectra and X-ray diffraction. The heat capacity (Cp) of NaCeF4 was measured by differential scanning calorimetry in the temperature range of 300–1093 K using the “step-method”. The Cp was fitted by an equation with a satisfactory result. Heat capacity was compared with that calculated from the Neumann-Kopp rule (NKR) and the deviations observed were consistent with the stability of the NaCeF4 compound.

Phase diagram and ternary bromides of the CsBr–LaBr3 binary system

The phase diagram of the CsBr–LaBr3 binary system was investigated by Calphad approach. The Gibbs energy functions of the ternary bromides Cs3LaBr6 and Cs2LaBr5 were evaluated using an independent polynomial to fit the experimental heat capacities. For the CsLa2Br7 ternary bromide, Neumann–Kopp rule was used. A two sub-lattice ionic solution model (Cs+)P: (Br−,LaBr6−3,LaBr3)Q was adopted to describe the liquid phase. The thermodynamic parameters for each phase in the considered system were optimized by using available experimental data from literature, which were used to calculate the phase diagram and thermodynamic properties of the CsBr–LaBr3 system.

Vapour pressure measurements on lanthanum polyphosphate and ultraphosphate by the transpiration method

The vapour pressure of P4O10 produced by decomposition of LaP3O9 and LaP5O14 (LaP3O9→LaPO4+1/2 P4O10; LaP5O14→LaP3O9+1/2 P4O10) was measured by the transpiration method over the temperature ranges (1123 to 1323)K and (923 to 1173)K, respectively. The standard thermodynamic quantities of LaP3O9 and LaP5O14 were derived by the use of second-law method and Neumann–Kopp rule. The derived thermodynamic quantities are consistent with the phase equilibria in the (La2O3+P2O5) system reported previously. There is a significant discrepancy between the present results and the literature data for LaP3O9 at T=298.15K, possibly due to the lack of accurate thermodynamic values of LaPO4 in the past. The thermal and chemical stabilities of LaP3O9 and LaP5O14 were analysed toward their proposed use as proton-conducting solid electrolytes in fuel cells. LaP3O9 is considered more promising from the viewpoint of proton conductivity and thermal stability. In the presence of water vapour or hydrogen gas, the decomposition of LaP3O9 and LaP5O14 would be accelerated significantly.

Thermodynamics of TmRhO3, Phase Equilibria, and Chemical Potentials in the System Tm–Rh–O

Phase equilibria in the system Tm–Rh–O at 1200 K is established by isothermal equilibration of selected compositions and phase identification after quenching to room temperature. Six intermetallic phases (Tm3Rh, Tm7Rh3, Tm5Rh3, Tm3Rh2, TmRh, TmRh2±δ) and a ternary oxide TmRhO3 are identified. Based on experimentally determined phase relations, a solid-state electrochemical cell is devised to measure the standard free energy of formation of orthorhombic perovskite TmRhO3 from cubic Tm2O3 and β-Rh2O3 in the temperature range from (900 to 1300) K. The results can be summarized as: ΔGf,ox° ± 104/J·mol–1 = −46474 + 3.925(T/K). Invoking the Neumann-Kopp rule, the standard enthalpy of formation of TmRhO3 from its constituent elements at 298.15 K is estimated as −1193.89 (± 2.86) kJ·mol–1. The standard entropy of TmRhO3 at 298.15 K is evaluated as 103.8 (± 1.6) J·mol–1·K–1. The oxygen potential-composition diagram and three-dimensional chemical potential diagram at 1200 K and temperature–composition diagrams at c...

Ambient Isobaric Heat Capacities, Cp,m, for Ionic Solids and Liquids: An Application of Volume-Based Thermodynamics (VBT)

Thermodynamic properties, such as standard entropy, among others, have been shown to correlate well with formula volume, V(m), thus permitting prediction of these properties on the basis of chemical formula and density alone, with no structural detail required. We have termed these procedures "volume-based thermodynamics" (VBT). We here extend these studies to ambient isobaric heat capacities, C(p,m), of a wide range of materials. We show that heat capacity is strongly linearly correlated with formula volume for large sets of minerals, for ionic solids in general, and for ionic liquids and that the results demonstrate that the Neumann-Kopp rule (additivity of heat capacity contributions per atom) is widely valid for ionic materials, but the smaller heat capacity contribution per unit volume for ionic liquids is noted and discussed. Using these correlations, it is possible to predict values of ambient (298 K) heat capacities quite simply. We also show that the heat capacity contribution of water molecules of crystallization is remarkably constant, at 41.3 ± 4.7 J K(-1) (mol of water)(-1), so that the heat capacities of various hydrates may be reliably estimated from the values of their chemical formula neighbors. This result complements similar observations that we have reported for other thermodynamic differences of hydrates.

Enthalpy increment and heat capacity of Pb 3Bi

Enthalpy increments of Pb 0.71Bi 0.29 compound and enthalpy change associated with peritectic decomposition reaction of the compound were determined using high temperature Calvet calorimeter. The heat capacity of the compound was determined in the temperature range 230–440 K using heat flow DSC. The enthalpy increment data was fitted into the following polynomial equations. Δ H 298.15 K T ( J/mol ) = − 6492.2 + 21.2775 T + 0.00919 T 2 − 199,341 T ( 298.15 – 457 K ) The enthalpy of decomposition reaction of the compound at the peritectic temperature, 457 K, was found to be 984 J/mol. The heat capacity of the compound determined using DSC was fitted into the following polynomial equation. C P ( J/mol K ) = 23.486 − 0.01482 T + 97,338.4 T 2 ( 228 – 457 K ) The heat capacity values obtained from DSC were in reasonably good agreement with the values calculated from enthalpy increment equation and both were slightly higher than the heat capacity values calculated using Neumann–Kopp's rule.

Density and related thermophysical properties of metastable liquid Ni–Cu–Fe ternary alloys

The densities of liquid Ni–Cu–Fe ternary alloy system were investigated by molecular dynamics method in combination with a MEAM ( Modified Embedded Atom Method) potential model. The temperature range is from 1000 to 2200 K, including both a broad superheating range and a large undercooled regime. The densities of six Ni 100 − 2 x Cu x Fe x alloys ( x = 0 , 10 , 20 , 30 , 40 , 50 ) decrease linearly with the rise of temperature at the superheated and undercooled states, and increase with the enhancement of Ni content. Among the simulated alloys, the densities of only liquid pure Ni and Ni 60Cu 20Fe 20 alloy are available in literatures, which are in good agreement with the calculated values. According to the relationship between the excessive volume and the alloy composition, it can be deduced that Ni 100 − 2 x Cu x Fe x alloys deviate from ideal solution. Moreover, a comparison was also performed between the calculated results and the approximated values by Neumann–Kopp's rule. Based on the obtained density data, the thermal expansion coefficients are also derived. It firstly decreases to a minimum value, and then displays a rise with the increase of Ni content.

Experimental study of phase transformation and specific heat of ternary zirconia-based oxides using differential scanning calorimetry

Phase transformation and specific heat of five ternary zirconia-based oxides were studied using differential scanning calorimetry (DSC). The ternary oxides were fabricated by doping 7YSZ (3.945 mol.% Y 2O 3–ZrO 2) with pentavalent oxides Ta 2O 5 and Nb 2O 5, trivalent oxides Sc 2O 3 and Yb 2O 3, and tetravalent oxide CeO 2. The addition of pentavalent oxides to 7YSZ increased the formation of monoclinic phase upon cooling in comparison to 7YSZ. The phase transformation from monoclinic phase to tetragonal phase took place in the temperature range of 500–700 °C. The incorporation of trivalent dopants effectively stabilized cubic phase to room temperature; the cubic phase, however, seemed to experience an order to disorder transition during subsequent heating and cooling cycle. Tetravalent oxide addition to 7YSZ increased the formation of tetragonal phase as compared to 7YSZ and no phase transformation was observed between 100 and 1400 °C. The specific heat values for the ternary oxides were measured from room temperature to 1000 °C. The experimentally determined values were compared to that calculated based on the constituent oxides in the system using Neumann–Kopp rule and the discrepancies were further discussed in this study.

High-temperature heat capacity and heat content of CaCu3Ti4O12 (CCTO)

The isobaric heat capacity of CaCu3Ti4O12 (CCTO) was measured from 300 to 1100 K using differential scanning calorimetry (DSC). The results were verified using drop calorimetric measurement of enthalpy increment (H-T - H-298.15) at T= 973 and 1073 K. The samples were dropped from room temperature into a calorimeter maintained at high temperature. The results show small negative deviation from Neumann-Kopp rule. The enthalpy and entropy increments are computed as a function of temperature and compared with values available in the literature. The results obtained in this study are consistent with available information on enthalpy and Gibbs energy of formation of CCTO.

Application of Neumann–Kopp rule for the estimation of heat capacity of mixed oxides

The empirical Neumann–Kopp rule (NKR) for the estimation of temperature dependence of heat capacity of mixed oxide is analyzed. NKR gives a reasonable estimate of C pm for most mixed oxides around room temperature, but at both low and high temperatures the accuracy of the estimate is substantially lowered. At very low temperatures, the validity of NKR is shown to be predominantly determined by the relation between the characteristic Debye and Einstein temperatures of a mixed oxide and its constituents. At high temperatures, the correlation between their molar volumes, volume expansion coefficients and compressibilities takes the dominance. In cases where the formation of a mixed oxide is not accompanied by any volume change, the difference between dilatation contributions to heat capacity of a mixed oxide and its constituents is exclusively negative. It turns out that in the high-temperature range, where the contribution of harmonic lattice vibrations approached the 3 NR limit, Δ ox C p assumes negative values. For more complex oxides whose heat capacity has contributions from terms such as magnetic ordering, electronic excitations, the applicability of NKR is only restricted to lattice and dilatation terms.

Thermodynamic assessment of EuBr 2 unary and LiBr–EuBr 2 and NaBr–EuBr 2 binary systems

As a basis for the design and development of molten salt mixtures, thermodynamic calculations of the phase diagrams and thermodynamic properties were carried out on the EuBr 2 unary and LiBr–EuBr 2 and NaBr–EuBr 2 binary systems over a wide temperature and composition range, respectively. The Gibbs energy of EuBr 2 was evaluated using an independent polynomial to fit the experimental heat capacity, the thermodynamic parameters for each phase in the LiBr–EuBr 2 and NaBr–EuBr 2 systems were optimized by using available experimental information on phase diagrams. A regular substitutional solution model for the liquid phase and Neumann–Kopp rule for the stoichiometric compound LiEu 2Br 5 were adopted to reproduce the experimental data with reasonable excess Gibbs energy. Comparisons between the calculated phase diagrams and thermodynamic quantities show that all reliable experimental information is satisfactorily accounted for by the present thermodynamic description. Some thermodynamic properties were predicted to check the suitability of the present calculation.

Thermophysical property of undercooled liquid binary alloy composed of metallic and semiconductor elements

The thermophysical properties of the liquid Ni–Si binary alloy system were investigated by the molecular dynamics method. The properties investigated include density, excessive volume, enthalpy, mixing enthalpy and specific heat at both superheated and undercooled states. It is found that the density decreases with an increase in the Si content, and so do the temperature coefficients. If the Si content is smaller than 30%, the density changes linearly with the temperature. If it is larger than 30%, the density is a quadratic function of the temperature. The simulated enthalpies of different composition alloys increase linearly with a rise in temperature. This indicates that the specific heats of Ni–Si alloys change little with temperature. The specific heat versus composition first decreases to a minimum value at 50% Si, then experiences a rise to a maximum value at 90% Si and finally falls again. According to the excessive volume and mixing enthalpy, it can be deduced that the Ni–Si alloy system seriously deviates from the ideal solution. Moreover, a comparison was also performed between the present results and the approximated values by the Neumann–Kopp rule. It reveals that this work provides reasonable data in a broad temperature range, especially for the metastable undercooled liquid state.

Heat capacities of several Al–Ni–Ti compounds

The heat capacities of several Al–Ni–Ti compounds were determined by drop calorimetry over the temperature range of 500–1500 K. A modified Einstein model and a two-parameter polynomial model provide reasonable representations of the experimental heat capacity data. The heat capacities, C p , using a two-parameter polynomial representation are as follows: Ni 0.5Ti 0.5, C p = 22.39 + 8.24 × 10 −3 T (J/(mol K)); Al 0.45Ni 0.5Ti 0.05, C p = 23.01 + 5.12 × 10 −3 T (J/(mol K)); Al 0.16Ni 0.74Ti 0.10, C p = 18.36 + 10.76 × 10 −3 T (J/(mol K)); and Al 0.25Ni 0.5Ti 0.25, C p = 25.38 + 1.088 × 10 −3 T (J/(mol K)). The experimental data are compared with the values derived from a thermodynamic database of Gibbs energy functions. The analysis shows that (1) either model is a good representation of the data; (2) it is not adequate to assume the Neumann-Kopp rule for the description of the heat capacities of Al 0.45Ni 0.5Ti 0.05, Al 0.16Ni 0.74Ti 0.10 and Al 0.25Ni 0.5Ti 0.25; (3) it is not appropriate to determine a compound C p from a thermodynamic database of Gibbs energy functions if the compound is modeled by using the Neumann-Kopp rule and any of its components undergoes melting in the temperature range of interest.

Thermodynamic characterization of liquid alloys with demixing tendency: Bi–Ga

Abstract The system Bi – Ga shows a pronounced miscibility gap in the liquid state ranging from 38.5 to 91.5 at.% Ga at 495 K with a maximum at 535 K at about 70 at.% Ga. Heat capacities of liquid Bi – Ga alloys were determined by heat flow differential scanning calorimetry over the entire composition range at temperatures between 520 and 600 K. In order to optimize the differential scanning calorimetry measurements in the liquid state, the sample masses were increased to 250–400 mg and the heating and cooling rates were reduced to 10 or 5 K min−1, respectively. The experimental results show a positive deviation from the Neumann-Kopp rule in the composition range of the miscibility gap. The results are in good agreement with theoretical considerations. Additionally, enthalpies of mixing were determined by drop calorimetry in the composition range from 30 to 100 at.% Ga at three different temperatures: 513, 573 and 673 K.