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... Elements fly apart - emitting "stuff" - whenever energy
conservation permits them to do so.
A nucleus in an excited state will decay to a lower state by
gamma emmission.
A nucleus will decay to another more stable nucleus of the
same A and charge different by one unit by beta emission.
A nucleus will split into nuclei of two or more different elements
if the split is energetically possible (think of the neutron and proton
as "elements" in this context).
You are really answering a different question - why are some
nuclei more stable than others. You should clarify the difference to
the students
Think of
a chain of students holding hands. The force is short range - two arm
lengths - but there is no limit to the length of a possible chain.
There is also a neutron "drip line" - possibly
because of the Pauli principle limiting the number of neutrons that
can be near the center. So having the right mix of n and p is crucial.