Hard-core potentials, including the Hard-Sphere (HS), Square-Well (SW), Sutherland (ST), Hard-core double Yukawa (HCDY), and Hard-Core Lennard-Jones (HCLJ) are of great importance in the liquid state theories. We show that we can derive analytical expressions for their radial distribution functions (RDF) at least at low density limit. So in the first step we derive a general analytical solution to the integral equations of liquid state theories at low-density limit for potentials with a hard core. For the specific case of the Lennard-Jones potential with a hard core, we have derived an analytical function for the radial distribution function at high temperature and low density. This function has been used to calculate the third virial coefficient for such a fluid exactly. In order to use the derived RDF for real fluids, one of the potential parameters is chosen in such a way that the RDF behaves more like that for a real fluid. We have then applied our procedure to a Lennard-Jones fluid and compared the results with simulation data. The agreement is good up to moderate densities. We have also applied our method to a real fluid, i.e., argon. We supposed that argon is an example of a Lennard-Jones fluid. The calculated results are in good agreement with experimental data for gaseous state of argon. Despite useful features of hard-core potentials, they have an unwilling behavior that must be improved. For instance, while these potential models correctly predict the behavior of the second virial coefficient [ B(T) ] at low temperature region, they show, however, a significant deviation from experimental data at high temperatures. This behavior is due to existence of a hard-core diameter in such potential models. Because the molecular diameter becomes smaller when temperature increases, we introduced a new expression for the variation of molecular diameter with temperature that incorporates this effect. The temperature dependence of the diameter was used in both ST and SW potential models to modify the second virial coefficients. Then we have shown that the experimental second virial coefficient fits into the modified ST and modified SW quite well for the entire temperature range for which experimental data are reported, including the inversion temperature. Then we have calculated the non-equilibrium properties of real fluids, i.e., viscosity, thermal conductivity, and self-diffusion coefficients. When these calculations were performed we found that the obtained results were in a closer agreement with experimental data, that is, there is no need to have two sets of potential parameters for the equilibrium and non-equilibrium properties of real fluids when the temperature dependence of the molecular parameter are taken into account, correctly.