We investigate in detail the computer simulation of argon adsorption on a graphite surface over a very wide range of temperature, from below the triple point to well above the critical point. Adsorption over such a wide temperature range has not been reported previously in the form of adsorption isotherms and enthalpy change during adsorption. The adsorption isotherms can be classified broadly into four categories: below the triple point, the isotherms show stepwise character (a strict layering mechanism) with 2D condensation; type II (according to the IUPAC classification) is followed by isotherms at temperatures above the triple point and below the critical point and a sharp spike is seen for isotherms in the neighborhood of the critical point; and finally the typical behavior of a maximum is observed for isotherms above the critical point. For the isosteric heat, the heat curve (plotted against loading) remains finite for subcritical conditions but is infinite (singularity) at the maximum in excess loading for supercritical adsorption. For the latter case, a better representation of the energy change is the use of the integral molecular enthalpy because this does not exhibit a singularity as in the case of isosteric heat, We compare the differential and integral molecular enthalpies for the subcritical and supercritical adsorptions.