Recent experimental evidence showed a strong correlation between the behavior of polymers under confinement and the presence of a layer irreversibly adsorbed onto the supporting substrate, hinting at the possibility to tailor the properties of ultrathin films by controlling the adsorption kinetics. At the state of the art, however, the study of physisorption of polymer melts is mainly limited to theory and simulations. To overcome this gap, we present the results of an extensive investigation of the kinetics of irreversible adsorption of entangled melts of polystyrene onto silicon oxide. We show that the process of chain pinning proceeds via a first order reaction mechanism, which slows down at large surface coverage, and the adsorbed amount scales with the predictions of reflected random walk. We propose an analytical form of the time evolution of the thickness of the adsorbed layer with two well-defined regimes: linear at short times and logarithmic at longer times, separated by a temperature independent crossover thickness and a molecular weight independent crossover time, in line with simulations and theory.