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More stable bonds have more negative (less) potential energies, which means more energy is required to break the bonds. Therefore bonds with greater potential energy (less negative) are less stable and require less energy to break.
If it takes more energy to break the bond, that means the bond is very stronger, and the attraction between the nuclei of the atoms is also very strong. According to Coulomb's law, stronger attractions mean a lower energy arrangement, which means more stability. Thus strong, high energy bonds have a lot of stability.
Stability is an indicator of how strong bonds are because the more stable something is, the more resistant it is to perturbations. So strong bonds should be able to withstand or require more energy in order to break them.
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