Among the various electrochemical conversion and storage systems, rechargeable Zn-air batteries have been considered as a promising candidate for future electric devices due to their high specific energy density. To improve the sluggish rate of oxygen reduction reaction (ORR) and oxygen evolution reaction (OER), which has been pointed out as a critical limitation of Zn-air batteries, pyrochlore oxides (A<sub>2</sub>B<sub>2</sub>O<sub>7</sub>) can be used as a highly efficient bi-functional electrocatalyst because of high charge transfer activity during ORR and OER. However, catalytic origin of them has not been fully explored due to complexities in structure and reaction mechanism. In this work, we investigated the reaction mechanisms for ORR and OER particularly in bismuth ruthenate pyrochlore oxide (Bi<sub>2</sub>Ru<sub>2</sub>O<sub>7</sub>) via density functional theory (DFT) calculation. Initially, we determined the energetically stable surface terminations for each low-index surface (i.e., (100), (110), and (111)). Next, we calculated the adsorption free energies of intermediates for oxygen-involving electrochemical reaction (i.e., OOH<sup>*</sup>, O<sup>*</sup>, and OH<sup>*</sup>). Finally, we constructed a 2D activity map of theoretical overpotentials (<i>??</i>) based on the adsorption free energies, which was used as a descriptor for bi-functional catalytic activity. Interestingly, the Bi and Ru sites were clearly separated in the activity map, induced by differences of binding strengths of reaction intermediates. The active sites with the minimum <i>??</i> were identified to be Bi site for ORR, and Ru site for OER, respectively. Based on the results, we theoretically demonstrated that the Bi<sub>2</sub>Ru<sub>2</sub>O<sub>7</sub> pyrochlore oxide is the excellent bi-functional electrocatalyst, which intrinsically contains different active sites separately working for ORR and OER, due to the distinct binding characteristics by each metal species