Optimizing the heat capacities of sphalerite phases as single system, or how nuclear physics can help physical chemistry

Abstract

The correct mathematical description of heat capacities C_p in a wide range of temperatures is still unsolved problem. A fragmental description of some phases is like a vision of one part of a large mosaic picture. A single description of C_p or other property of a phase of any isostructural series does not allow one to see the integrity of the entire ensemble. We propose a special mathematical model to describe C_p in a wide temperature range for a whole large class of isostructural sphalerite phases. In the proposed model, it is believed that an ideal crystal does not have any foreign inclusions, defects, or dislocations. The group IV elements (Si, Ge, α-Sn and diamond-like Pb) were taken as the basis, with flerovium (¹¹⁴Fl) closing this group. There should be no other elements in this group according to the fine structure constant (α) (or the Sommerfeld constant). As a consequence, the limiting value of the heat capacities of phases with a sphalerite structure falls on the element 114 (¹¹⁴Fl) and has a value of C_p = 30.5 ± 0.3 J · mol-at⁻¹ · K⁻¹. This value was obtained as a maximal virtual point C_p of the last element (¹¹⁴Fl) of group IV and corresponds to Ln (C_p/R) = 1.30 ± 0.01 for the isotherms ln (C_p/R) vs Ln(N), where N is an atomic number of an element of group IV or the sum of the atomic numbers of A^IIIB^V or A^IIB^VI compounds per mole-atom. The common point of heat capacity attributable to flerovium is obtained from the linear equations С_р/R vs Ln(N) at low temperatures from 25 to 35K. For only pure elements of group IV (Si, Ge, α-Sn and diamond-like Pb), flerovium closes this group, and there are no other elements behind it, according to α. The maximum heat capacity of flerovium can be taken as 30.5 J·mol-at⁻¹·K⁻¹ with an accuracy of 1%. As the temperature decreases, this value slowly decreases (within 1%), and then, when it approaches 0 K, it drops sharply to 0 J·mol-at⁻¹·K⁻¹. To describe the set of the isostructural experimental data C_p(T) for diamond-like phases in solid state as a whole system, here we used a special multi-parameter family of functions. For each substance, the parameters are found by minimizing the discrepancy between the theoretical dependence C_p(T) and corresponding experimental data. The dependence of the heat capacities for elements of group IV (Si, Ge, α-Sn, diamond-like Pb, Fl) at fixed temperatures on Ln(N), where N is the atomic number or the demi sum of the atomic numbers of phases A^IIB^VI or A^IIIB^V. In this case, either a break point or an inflection point attributable to germanium is observed for parameter dependencies on Ln(N).