The validity of the hard-sphere (HS) model in determining the self-diffusion coefficient of simple fluids is studied using the Lennard-Jones (U) fluid as a test fluid. The latest simulation results on the LJ fluid are used along with the various perturbation theories to assess the HS model separately in the liquid, gas, and supercritical regions as a function of both density and temperature. Apart from the commonly used methods of estimating the effective hard-sphere diameter (EHSD), the Gibbs-Bogoliubov criterion, which provides an EHSD that is more representative of the intermolecular interactions, is used. This criterion gives a vastly different assessment of the HS model compared to other methods in the supercritical and liquid regions. All methods result in considerable errors in the low-density dilute gas region at lower temperatures. Next, the diffusivity of carbon dioxide is studied. Nine different sets of U potential parameters and three non-LJ potentials are considered. The U potential based on the critical constants of carbon dioxide yields diffusivities closest to the experiment.