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More specifically, how and why is there energy stored in the strong attractive bonds that hold multiple atoms together? I am struggling to understand this as my understanding is that energy is released when the atoms bond, so why is there potential energy stored in the bonds?

But also, I feel like this is crucial in understanding this, is 'potential energy' only the ability to release energy due to its position etc, or is it also the ability to gain energy? If the latter is correct, then does that mean atoms have the potential to gain energy when energy is applied to the atom to break the bond?

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It's not. Things are bound together when, bonded, they contain less energy than when separated.

Put another way, energy is released to form the bonds and energy has to be added to break them. The Energy that gets released when the bonds are formed was stored in the electromagnetic fields inside of the atom. When the bond is formed, the electrons and protons assume a configuration that has an, overall, less intense combination of electric and magnetic fields inside of the bonded atoms than when they were separate. The released energy travels off in the form of electromagnetic waves or as faster motion of atoms and molecules.

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All bound systems can be to first order approximated to a potential well, classically and quantum mechanically.

Water in a reservoir has high potential gravitational energy, potential in this case means "it is possible to get this much energy if the water reaches the ground"

Quantum mechanically the meanings do not change , and chemical bonds are in the realm of quantum mechanics, but the reasoning for the definition of potential is the same: there exists a lower energy state, a ground state.

bonds

Atoms and molecules release energy and settle at the lowest ground state of the potential in question.

'potential energy' only the ability to release energy due to its position etc,

Yes, as shown above. Once bonded, energy has to be supplied so that the "free" state (ionization state) can be achieved.

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