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I was solving a problem in organic chemistry, where I had to find the resonance energy of a compound with some information which was given in the question itself. Basically, what I had to do was, to sum up all the energies from the information given and then subtract it with the experimental value of energy given. My question was that if the law of conservation of energy was to be followed while solving the problem then where was the energy which I obtained as a difference of the experimental and theoretical data used? And if it is just a calculation of experimental error, then why was it called as resonance energy, was it related to its name in anyway.

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What I understand from your question is that you were given different bond energies of the various bonds constituting a compound which was showing some resonance and also you were given the experimental value of the energy of that compound.

Now as a compound is formed when different atoms form bonds together so, the total energy of that compound should be equal to the sum of all the bond energies of bonds between different atoms of that compound but this is not the case (most of the time in the real world).

Every compound tries to go to the lowest energy as possible and one way to do so is by doing resonance.

Resonance is the phenomenon of delocalisation of bonding electrons of a bond between many atoms of a compound rather than just two atoms.

By doing resonance, the electrons in a compound are more evenly distributed and delocalised resulting in more stability of the compound and thus a lower energy. So, the actual value of the energy of a compound is lower than just the sum of bond energies of all its bonds and the decrement in the value is equal to the energy it released in the form of resonance energy. Yes! It is released in the surroundings.

I hope it helped.

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As a guide if you are familiar with the quantum model for a particle in a box you will know that the levels becomes lower in energy as the box gets larger. The same is true for a particle on a ring as the ring's radius increases. So, if you have three isolated double bonds, vs. the same bonds with delocalised energy (double - single - double -single etc., and you may call this resonance if you wish) then you can see that the latter has lower energy.

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