The transition states of the elementary reactions for the dissociation of methane on the ruthenium (0001) surface have been investigated with DFT periodic calculations and the nudged elastic band method (NEB) for 25% coverages. The calculated barriers are 85 kJ mol(-1) for methane decomposition, 49 kJ mol(-1) for methyl decomposition, and 16 kJ mol(-1) for methylene decomposition, respectively. The decomposition of CHads requires the highest activation energy from the series with 108 kJ mol(-1). Discussion concerning chemical bonding aspects of the transition states structures is provided for.