Thermodynamic properties of monoclinic β-sulfur Sβ under high pressure up to 20 Gpa have been studied, where Orthorhombic α-sulfur Sα changes to Sβ at 94.4℃. The high pressure technology used to tackle the crystallization volume of sulfur Sβ without changes in the chemical properties. Sulfur Sβ can be used after the effect of high pressure in the development of the medicine used to treat scabies. The present study is performed to calculate the effects of high pressure on some important physical properties of the material such as (bulk modulus (B), compression volume ratio (VP/Vo), Grüneisen parameter ), Debye temperature (θD) and phonon frequency spectrum (pfs). Three different equations of state EOS's (Birch-Murnaghan EOS, Vinet EOS and modified Lennard-Jones EOS) were implemented to analyze pressure-volume relationship and then combining calculated data with various expressions of volume dependence of the other thermodynamic properties that are; Bulk modulus, Grüneisen parameter, Debye temperature and phonon frequency spectrum. The behavior of any of these concepts were described in different figures. It was found that, relative volume, and Grüneisen parameter were decreased with high pressure, while the other considered parameters; Bulk modulus, Debye temperature and phonon frequency spectrum were expected to increase as a high pressure is applied.

The pressure equation of state of Titanium nitrate TiN was investigated in the current work using two equations of state (EOS) from the literature, including the Barden EOS and the Birch-Murnaghan EOS, whereas the Bardeen EOS, which is based on interstellar atomic potentials, and the Birch-Murnaghan EOS, which is based on the solid mechanics notion of finite strain. The EOSs were processed to identify the impacts of high pressure on the bulk modulus B, Debye temperature θD and lattice constant a, which are characterizations of TiN. Ultimately, a fair comparison of the current findings with the first principle approximation and the generalized gradient approximation approach was conducted, and a perfect agreement was found. It was demonstrated that TiN EOS can be used to calibrate high pressure for chemical compound TiN .