The coverage-dependent phase behavior of molecular films of alcohols (CH3(CH2)(n-2)CH2OH, denoted as CnOH) on mercury was studied for chain lengths 8 less than or equal to n less than or equal to 28, using surface tensiometry and surface specific X-ray methods. Phases with surface-normal-oriented molecules are found at high coverage, showing the CS, S, and LS phases found also on water. Phases comprising surface parallel molecules, which do not exist on water, are found here at low coverage. For the lowest coverage a two-dimensional gas phase is found, followed, upon increasing the coverage, by an n-dependent sequence of condensed phases of up to four layers of surface-parallel molecules before converting to the surface-normal phases. In contrast with the surface-normal phases, all of the surface-parallel phases are found to lack long-range order in the surface-parallel direction. Adsorption energies are derived from the phase diagram for the alkyl chain and the alcohol headgroup.