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Nuclear reactions that occur in nuclear bombs and nuclear reactors are not chemical reactions, strictly speaking. Chemical reactions involve the formation and breaking of bonds through the sharing of electrons which are outside of the nucleus, while nuclear reactions involve the formation breaking of atomic nuclei. Keep that distinction in mind as you read this answer.

Hydrogen bombs fuse two nuclei to form a new nucleus. Atomic nuclei are composed of two types of smaller parts, called protons and neutrons. A hydrogen atom composed of one proton and one neutron are smashed into another hydrogen atom composed of one proton and two neutrons. A new atom is formed, a helium atom with two protons and two neutrons. The extra neutron flies away as well.

It turns out that this new nucleus and the extra neutron have less energy than the two starting hydrogen atoms. The extra energy is released as heat and light. This amount of energy is rather large (as far as the atomic scale is concerned), and when a very large number of these nuclear reactions occur, as in a hydrogen bomb, the total amount of energy can be enormous.

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