How do I explain and show the proper chemical equation for the mechanisms behind the energy source nuclear fission?  

To properly explain the equations for nuclear fission reactions, all you need to know is what high-molecular weight molecule is involved. Fission reactions involve a neutron (or multiple neutrons) colliding with large molecules, which produces smaller molecules and energy. The most famous nuclear reaction is the collision of a neutron with U-235, which produces energy, Ba-144, Kr-86, and three new neutrons.

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Nuclear fission is the process of bombarding high-density, high-molecular weight atoms with neutrons in order to cause them to split, thereby releasing two lower-molecular weight atoms and a tremendous amount of energy. These kinds of reactions usually involve isotopes, or atoms that contain a different number of neutrons than protons in the nucleus. Thus, the proper chemical equation will usually represent a neutron colliding with a high-molecular weight molecule as reactants and the smaller-molecular weight molecules as products, as well as the energy release.

Additionally, the breakup of a large atom not only produces atoms of smaller molecular weight, but also more free neutrons. These neutrons, once they are released from the nucleus of the larger-molecular weight molecule, are then free to collide with other molecules, initiating what is called a nuclear chain reaction.

The most common type of fission reaction is when a neutron collides with a molecule of Uranium-235. The first product of this reaction is an extremely short-lived molecule of Uranium-236 (which has just absorbed the incoming neutron). This molecule, however, is extremely unstable and quickly breaks down into three separate products: Barium-144, Krypton-86, and three neutrons.

The reaction also produces gamma-ray radiation and energy. The energy release in Joules for nuclear reactions can often be looked up in a chemical table.

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