The osmotic pressures of polymer solutions are related to their corresponding osmotic virial coefficients. There are several experimental methods to determine the osmotic virial coefficients. However, these experimental virial coefficients are restricted to the experimental conditions. An alternative way of determining the osmotic virial coefficient is with the help of statistical thermodynamics, and unlike experimental methods, because of its theoretical basis, it is not restricted to limited ranges of experimental conditions. The osmotic virial coefficients can be used to calculate the different properties of polymers and polymer solutions, such as the average molecular weight of the polymer (M) and the theta temperature. In this work, for the first time, the second osmotic virial coefficients of a number of polymeric solutions are calculated using the Kihara and Exponential-6 intermolecular potential functions. Furthermore, the hard-core Exponential-6 model, that we have proposed, is also investigated. Additionally, the molecular weights and theta temperatures of these solutions are determined using the calculated second virial coefficients and are compared to literature data. The results show that the recent hard-core Exponential-6 potential model is more accurate than the Kihara and Exponential-6 models and is applicable for calculating other properties of the polymers and polymer solutions.