“…The integral equations theory is a significant theory for discussing thermodynamics of multiple spectrum of materials such as soft materials and simple fluids. For instance, Zhou et al (2004) applied the integral equations theory for finding the depletion potential for the colloidal particle, Aguirre-Manzo and González-Mozuelos (2021) discuss the charged colloids with high charges with the salt in suspensions using the theory of the integral equations, Pérez-Molina et al (2021) used the chaotic data with the theory of the integral equations for the estimation of the virial coefficients of the equation of state, Kalyuzhnyi et al (2021) applied the theory of the integral equations with the numerical analysis techniques for the study of the mixtures of colloids, Hashimoto et al (2019) use the theory of the integral equations with the atomic force microscope data for the determination of the number density distribution of colloidal particles on a substrate, Filippov et al (2019) used the theory of the integral equations for the dusty plasma mixtures, Munaò and Saija (2019) studied the hertzian spheres in case of the low temperatures using the theory of the integral equations in addition to the Monte-Carlo simulation, Herrera (2018) found some structural and thermodynamic properties of fluids described by hard sphere and Yukawa potential using the theory of the integral equations, Arauz-Lara showed some application of the integral equations theory to the colloidal fluids, Pizio et al (2011) discussed a model for the simple fluids using the second order of the integral equations theory, Lukšič et al (2011) applied the integral equations theory for the mixtures of the electrolytes and non-charged hard spheres for purpose of the determination of the structural properties and thermodynamics of these mixtures, Fukudome et al (2014) showed a new formula for the direct correlation function for the hard sphere fluids, Wu et al (2014) calculated the static structure factor for the charged spheres using the integral equations theory, Lomba et al (2015) used the three dimensional integral equations approximation for discussing of fluids with confinement and applied in the case of the argon in zeolites, Miyata and Miyazaki (2016) discussed the one component fluids interacting via Lennard-Jones interaction for the study of the temperature derivative of the radial distribution function using the integral equations theory, Melnyk et al (2016) found the structure factor of the fluids of the hard core type interacting via short range Yukawa potential using the integral equations theory and the simulation methods, Al-Raeei (2022a) derived a state equation for London interaction. Also, Al-Raeei (2021) applied the theory for the study of some soft materials, Zhou (2010) discussed the fluids with the honeycomb interaction with the integral equations theory.…”