Vibrational Entropy Change Research Articles

Calorimetric study of the stability of spinelloids in the system NiAl2O4-Ni2SiO4

Enthalpies of solution in molten 2 PbO · B2O3 at 974 K were measured for four spinelloids, phases I (0.75 NiAl2O4 · 0.25 Ni2SiO4), II (0.60 NiAl2O4 · 0.40 Ni2SiO4), III and IV (0.50 NiAl2O4 · 0.50 Ni2SiO4) in the system NiAl2O4 · Ni2SiO4. The enthalpies (in cal per 4-oxygen mol) of formation from NiAl2O4 and Ni2SiO4 spinels are: phase I, 945±366; phase II, 1072±360; phase III, 2253±390; phase IV, 3565±544. Using these enthalpy data in combination with phase relations at high pressure at 1373 K, positive entropies of formation of the spinelloids from NiAl2O4 and Ni2SiO4 spinels were estimated (in cal mol−1 K−1): phase I, 1.2; phase II, 1.5; phase III, 2.0–2.3; phase IV, 3.0–3.1. The thermochemical data obtained above suggest that the spinelloids are “entropy-stabilized” phases with partially disordered cation distributions. The configurational entropies of the spinelloids were calculated based on the observed cation distribution in each spinelloid phase. The positive entropies of formation of the spinelloids from the spinel endmembers are due primarily to the configurational entropies although small positive vibrational entropy changes may also exist.

The entropy of formation of an anion frenkel defect in calcium fluoride

Abstract The vibrational entropy change at constant volume that is associated with the formation of anion Frenkel defect in calcium fluoride has been calculated in the quasiharmonic approximation by finding the vibrational frequencies at q = 0 of an electrically neutral 32-molecule “supercell” containing both an anion vacancy and an anion interstitial. The supercell is of a size and shape such that every ion that is a first or second neighbour to a defect can be given the position it occupies in the displaced, defect lattice. From these results, and the calculated energy of formation of a Frenkel defect at constant volume, we have calculated the enthalpy, entropy and volume changes associated with Frenkel defect formation in calcium fluoride at constant pressure.

Numerical calculations of the change of the vibrational entropy due to disordering of a one- and two-dimensional ordered binary alloy

Abstract The change of the vibrational entropy at high temperature due to the disordering of an ordered alloy has been calculated for the linear chain and the square lattice within a harmonic model of nearest-neighbour interactions. For the two-dimensional lattice a step-wise numerical procedure was used. The change in vibrational entropy turned out to be of a similar magnitude to the change in configurational entropy.

The Vibrational Entropy Change Associated with the Disordering of a Two‐Dimensional Binary Alloy Using a Model of Nearest‐ and Next‐Nearest‐Neighbour Interactions

AbstractIn a model of harmonic central forces, acting between nearest and next‐nearest neighbours the change of vibrational entropy due to disordering of a completely ordered binary alloy is calculated numerically. The crystal structure is the two‐dimensional square lattice. The temperature is well above the Debye temperature. A comparison is made with results obtained previously using a different model. It turned out that the values of the entropy change are not too much model dependent in the examples investigated. Again it appears that the change of the vibrational entropy is of a similar magnitude as that of the configurational entropy.

Thermodynamical analysis of optimal recombination centers in thyristors

The main criterion for an optimal recombination center in thyristors has been reformulated, starting from a basic thermodynamical level. From an extended grand canonical ensemble, statistical expressions for the thermal emission rates of an impurity atom are obtained in terms of changes in enthalpy, electronic and vibrational entropy. The ratio between lifetimes at high and low injection levels is evaluated for single and double level recombination centers and calculated for different enthalpy positions and entropy changes of the center and for different resistivities of the thyristor middle region. It is shown that the expected changes in entropy, when a charge carrier is emitted or captured by the center influences the optimal enthalpy position in silicon by as much as 0.1 eV. Also, a complete change in the influence of injection level on lifetime at standard resistivities and temperature for high power devices is noted. The optimization criterion is compared with experimental data pertinent to thyristor optimization. The paper demonstrates the necessity of making a profound thermodynamical analysis of the charge carrier traffic at a recombination center when using experimental data for deep impurities in the optimization of thyristor lifetime ratings.

Investigation of the phase diagram KClBaCl2 in the region of solid solutions with the base of KCl

AbstractThis paper is concerned with the determination of the lines of maximum solubility of the phase diagram for the system KClBaCl2 in the region of solid solutions with the base of KCl by the method of light scattering. The results are discussed in terms of the theory of dilute solutions of solid electrolytes. The enthalpy of solution Ψ = 0.64 eV and the change of vibrational entropy ΔS/K = 1.9 are defined also.

Erratum: “Lattice Softening, Phase Stability and Elastic Anomaly of the β-Au-Cu-Zn Alloys”

The β-Au x Cu 1- x Zn alloys transform martensitically from a Heusler-type ordered structure (b. c. c.) into an orthorhombic crystal at lower temperatures. The phase stability associated with the transformation was investigated by an examination of the elastic constants in the Heusler phase. The shear constant ( C 11 – C 12 )/2 approaches zero near the martensite temperature, and its temperature coefficient is positive in the region above martensite temperature. Thus the transformation in these alloys proved itself to be deeply connected with a lattice softening and the elastic anisotropy. The free energy difference between the Heusler and the martensite phases mainly comes from the change in vibrational entropy, which can explain the thermoelastic property of the transformation. Pseudoelastic properties including ferroelasticity and the memory effect of these alloys were also investigated from their strain-stress diagrams. The properties have a close relation to the large elastic anisotropy and the latti...

Lattice Softening, Phase Stability and Elastic Anomaly of the β-Au-Cu-Zn Alloys

The β-Au x Cu 1- x Zn alloys transform martensitically from a Heusler-type ordered structure (b. c. c.) into an orthorhombic crystal at lower temperatures. The phase stability associated with the transformation was investigated by an examination of the elastic constants in the Heusler phase. The shear constant ( C 11 – C 12 )/2 approaches zero near the martensite temperature, and its temperature coefficient is positive in the region above martensite temperature. Thus the transformation in these alloys proved itself to be deeply connected with a lattice softening and the elastic anisotropy. The free energy difference between the Heusler and the martensite phases mainly comes from the change in vibrational entropy, which can explain the thermoelastic property of the transformation. Pseudoelastic properties including ferroelasticity and the memory effect of these alloys were also investigated from their strain-stress diagrams. The properties have a close relation to the large elastic anisotropy and the latti...

Vibrational entropies of hydrogen in palladium-silver-hydrogen alloys by the isotopic solubility ratio method

The solubilities of hydrogen and deuterium in Pd−Ag alloys containing 10, 20, 30 and 40 wt. % Ag were measured at 125°C by the Sieverts' technique using Pd black as the transference catalyst. The results obtained were analysed in terms of the isotopic solubility ratio theory. In the Pd−10Ag−H, Pd−20Ag−H and Pd−30Ag−H alloy systems, there was no significant change in the partial vibrational entropy of hydrogen S H/ V in the range of hydrogen contents H/M from 0·025 to 0·325, nor was there any marked difference between these systems. However in the Pd−40Ag−H system, S H V decreased markedly with increasing hydrogen concentration in the range of H/M from 0·050 to 0·150.

Intermolecular forces in gaseous ammonia and in ammonia – nonpolar gas mixtures

Intermolecular potential parameters for ammonia have been determined for the Stockmayer–Kihara function using experimental second virial coefficient, diffusivity and viscosity data of binary mixtures with argon, methane, nitrogen, and oxygen. The parameters U0/k = 215 °K, σ(core-to-core) = 2.70 Å and a* = 0.2 reproduce essentially all of the data nearly within experimental error and accurately reproduce pure ammonia transport properties. Upon considering both physical and chemical contributions to the second virial coefficient of ammonia, the apparent standard-state hydrogen-bond enthalpy for chemical dimerization is −3.2 kcal/mole while the vibrational entropy change is 20.2 cal/mole °K, indicating weak association.

Elastic and Anelastic Behavior during the Martensitic Transformation in Indium-Thallium Alloys

Measurement of internal friction at strain amplitudes <10−5 in In-rich In–Tl alloys has shown an amplitude-dependent peak associated with the fcc→fct martensitic transformation. The peak is thermally activated when the transformation occurs near the melting point Tm, but not at lower temperatures. It is suggested that the peaks originate from the stress-assisted motion of twin boundaries that are known to spontaneously coalesce immediately below the transformation temperature. The activation energy associated with the transformation peak near Tm is 10–13 kcal/mole higher than for the diffusion of Tl in In. A mechanism is proposed for this relaxation, invoking an interaction between twin boundaries and solute, whereby reordering of solute is effected in differently orientated tetragonal regions by twin-boundary motion. Shear modulus measurements indicated abrupt changes at the transformation temperature and positive temperature coefficients of modulus in the high-temperature fcc phase, implying structural instability and an ``entropy-elastic'' state. At 0.9 Tm, about 60% of the elasticity in a 19.5% Tl–In alloy arises from a change in vibrational entropy with extension. The results are consistent with the known crystallography of the transformation.

The vibrational entropy of Al-Ag solid solutions from X-ray diffraction data

The high excess entropy on solid solution formation in Al-Ag system was found by Hillert, Averbach, and Cohen from electrochemical measurements. This cannot be accounted for by strain entropy component due to the absence of atomic size effect in these alloys. The X-ray investigation of thermal vibrations of atoms in equilibrium solid solutions of Ag in Al was therefore carried out to prove whether the excess entropy was of vibrational origin. The mean-square amplitudes of thermal vibrations were determined from the measurements of integrated intensities on X-ray diffraction photographs of polycrystalline samples with silver contents from 0 to 10 at. % taken at a temperature of 625°C. The expression was derived enabling the calculation of mean-square displacements of atoms having different proportions of like and unlike nearest neighbours from the measured effective amplitude. It was found that the mean-square displacements of both the Al and the Ag atoms increase with increasing the number of unlike neighbours in their first coordination shell. An expression for vibrational entropy was derived taking into account all these vibrational states of atoms. Using this expression, the quantitative agreement was found between the excess entropy and the vibrational entropy determined from the present X-ray measurements. The excess entropy can be thus explained satisfactorily as a change of vibrational entropy due to the change of the vibrational state of atoms caused by new interatomic interactions between the solvent and solute atoms on solid solution formation.

The determination of the vibrational entropies of solute atoms in dilute copper solid solutions by the measurement of vapour pressures

An effusion method has been used to investigate the equilibrium between dilute copper based solid solutions and their vapours over a large range of temperature. Neutron irradiated alloys were used and a gamma-ray spectrometer enabled measurements on very dilute alloys to be carried out. The change of vibrational entropy S + M − S 0 v occurring when a solvent atom is replaced by a solute atom and the corresponding enthalpy change H + u − H 0 v were derived from the experimental data. The results are given in the table together with the values of H + u , the energy of a solute atom in the solution with respect to a solute atom at rest in a vacuum. System H u + − H v 0 kcal mole H u + kcal mole S u + − S v 0 k Cu-0.1 At. %Ag 26.6 ± 0.69 −45.9 ± 0.69 4.71 ± 0.29 Cu-0.5 At. %Au 2.01 ± 0.29 −70.5 ± 0.29 7.07 ± 0.12 Cu-2.5 At. %Au 1.84 ± 0.26 −70.7 ± 0.26 6.85 ± 0.11 To explain the very large increases in vibrational entropy two models have been discussed. In the first, atomic force constants were used to estimate the change in vibrational frequency with solute diameter for a cluster of 13 atoms. In the second model elastic theory was used to calculate the change in entropy due to the shear and dilational strains around a solute atom. The experimental values of S + u - S 0 v lie between those derived from the two models. Experimental values of the vibrational entropies deduced by other workers from solvus curves and e.m.f. measurements are also compared with the calculated values.

Theoretical calculation of distribution coefficients of impurities in germanium and silicon, heats of solid solution

The distribution coefficients of Groups III, IV and V impurities in germanium and silicon were calculated, and very satisfactory agreement with experimental values was obtained. The method of calculation consisted in estimating the change in energy involved in transferring the impurity atoms from a crystalline reservoir to the solid and molten host material respectively. The binding energy of an impurity in the host lattice and the strain energy which it introduces were estimated. The change in vibrational entropy is believed to be negligible.

On Simple Binary Solid Solutions

The effect of difference in atomic volume between two components of a simple binary solid solution on the phase equilibrium curves is investigated by calculating the energy of deformation required to replace a smaller atom by a larger atom, and vice versa. The strain energy is introduced to the free-energy function derived by the quasi-chemical technique of Guggenheim and leads to expressions similar to those of Hildebrand for unsymmetrical segregation. In addition, the Mie-Gruneisen equation is used to estimate the effect of vibrational entropy changes. It is found that the effect of such changes has a smaller effect on phase equilibria in general than the energy changes. Finally, the results are used to justify certain empirical thermodynamic expressions.