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The kinetic activation barriers and the thermodynamic free energy changes for the probable elementary reaction steps of oxygen reduction reaction(ORR)are calculated by the first principles methods to defuse the debate that whether the 2e-or 4e-pathway dominates the ORR on the CoN4 embedded graphene(Co-N4-gra).It is found that the Co-N4-gra can promote the ORR along a 4epathway and finally generate two H2O by successive hydrogenation reactions,and the reduction of OOH into O and H2O with the largest barrier of 0.69 eV is suggested as the rate-determining step(RDS)in kinetics.The thermodynamics results show that the elementary steps of ORR along the 4epathway are downhill at low electrode potential for U < 0.58 V.The last step of reduction of OH into H2O with the largest ΔG value(-0.58 eV)functions as the thermodynamic RDS of the 4e-pathway.The large/small energy barrier and small/large thermodynamic driving force for the generation/dissociation of HOOH indicate also that the 2e-pathway is less favorable than the 4epathway for ORR on Co-N4-gra.