Schottky contributions in chemical thermodynamics

Abstract

Cryogenic heat-capacity determinations provide a useful tool for the determination of the energetic spectrum of condensed phases and also reveal information on their discrete electronic level structures as well. We have been interested in applying these techniques to actinide elements and have in recent months been working up the techniques to unravel the corresponding data for the lanthanide compounds--where opposite trends in cationic masses and molar volumes provide an opportunity to test theories useful for the resolution of excess heat capacity from lattice contributions.As an important aspect of heat capacities---especially of compounds with d andfelectrons--the Schottky contribution deserves to be much better known--by chemists, by physicists, and by students of thermodynamics. These remarks are designed to further that goal. Schottky's "Orphaned" contributionWerner Schottky first showed in 1922 [1] that a contribution to the energetics--spectroscopic or thermophysical---of a substance could be expected from excited electronic levels. The splitting of the ground-level term by the crystalline electric field also gives rise to the appearance of such a "Schottky anomaly" or "Schottky contribution" either in the heat capacity or in other thermophysical properties. The manifestation of this contribution to the heat capacity is a characteristic bellshaped curve skewed out on the high-temperature side. The temperature of the peak is related to the energetic separation of the levels and the maximum height is determined by the ratio of the degeneracies. Moreover, the total entropy under the curve is also related to the number of levels and the degeneracies involved. When the electronic energy increments are sufficiently small, the Schottky anomaly occurs at so low a temperature that, in comparison to the lattice contribution, the Schottky contribution stands out clearly. When energy levels are sufficiently spread so as to cause the occurrence of the contribution near or above ambient temperature, the maximum is typically so spread out as to be unrecognized unless the heat capacities of isostructural substances with nearly identical atomic masses are used for John Wiley & Sons, Limited, Chichester Akad~miai Kiadr, Budapest

Density functional calculations of the enthalpies of formation of rare-earth orthophosphates

“…If LaPO 4 (m) is ignored, the difference between calculated and measured heats of formation for the monazite phases fall on a systematic trend with a negative slope (increasing atomic number implies increasingly negative deviation from experiment; in other words, as atomic number increases, the calculated heats of formation become less and less exothermic relative to the measured heats of formation). There will also be Schottky-type contributions for some of the phases (Westrum, 1985), but these cannot be responsible for the overall 40 kJ/mol differential between the measured and calculated enthalpies, as they should be close to zero for compounds with no crystal field stabilization (Y, Sc, La, Gd).…”

“…There will also be Schottky-type contributions for some of the phases (Westrum, 1985), but these cannot be responsible for the overall 40 kJ/mol differential between the measured and calculated enthalpies, as they should be close to zero for compounds with no crystal field stabilization (Y, Sc, La, Gd).…”

Heat capacities and derived thermodynamic properties of lithium, sodium, and potassium disilicates from T=(5 to 350)K in both vitreous and crystalline states

“…The adjustment to stoichiometric end member composition was made by utilizing the heat capacities of vitreous silica [13] and of a-quartz [13] for the vitreous and crystalline materials, respectively. The Schottky-function-like [14] deviation between the heat capacity of the vitreous phase and that of the corresponding crystal for each disilicate (M 2 O AE 2SiO 2 ; M = Li, Na, K) is shown in figure 3. The heat capacity of vitreous and crystalline Li 2 O AE 2SiO 2 has been measured between T = 6 K and T = 304 K by Bennington et al [4] and the percent deviation of the measured sample with reference to our smoothed curve is shown in figure 4.…”

“…The relative differences in total C p value for the lithium disilicates at T = 6.5 K is 2.5 compared with 1.25 at T = 103 K; for sodium disilicates at T = 5.5 K is 2.25 compared with 1.1 at T = 101 K; for potassium disilicates at T = 6.6 K is 2.2 compared with 0.99 at T = 100 K. These values are consistent with observations for other vitreous-crystalline systems. Westrum has demonstrated that the difference in the C p vs. T curves of a vitreous phase and the corresponding crystalline phase is a Schottky function [14,16]. The difference curves, [C p (vit.)…”

Thermal expansion and heat capacities of holmium and erbium orthotantalates ceramics

“…To perform this simulation, temperature dependences of thermodynamic functions of the substances involved in the process are required, as well as the reliable values of enthalpy of formation. Since the heat capacity of lanthanide compounds (with the exception of diamagnetic lanthanum and lutetium) contains an anomalous contribution due to the interaction of the 4f‐electron shell with the crystal field (Schottky anomaly), 22 estimation of their heat capacity by the Neumann‐Kopp rule can have a significant error, especially if the structural types do not match. Due to this reason there is a need for experimental measurement of the heat capacity in a wide temperature range.…”