Here's a few things that could happen, depending on the atom.
Light atoms would actually absorb energy
Atoms lighter than Iron 56 tend to require more energy to split than they would give off when split. Atoms on this side actually need to fuse in order to get any energy out of them, with the fusion of hydrogen into helium being the main source of energy for stars like our sun. So, splitting light atoms would end up being a loss of energy!
Most heavy atoms give off energy, then stop
Atoms heavier than Iron 56 will give off some energy when they split, but after that, they stop. Some elements, such as thorium, can give off enough energy to make nuclear power, but must be actively "fed" neutrons in order to work, and will stop if a source of neutrons used to split the thorium are cut off. This could be used for safer nuclear power, though the inability to build nuclear bombs from thorium had led to far less interest from most of the world's governments. Aside from thorium, however, most "heavy" elements do not even give off enough energy for nuclear power.
A few heavy atoms "Go Nuclear!"
A handful of elements (specifically isotopes) are able to both release energy when they split, and release the particles needed to split like atoms. This allows the process to feed itself, and grow exponentially. Uranium-235 is a well known example; put enough U-235 in one place (known as critical mass), and BOOM! Nuclear explosion!