Question

When the first nuclear bomb was detonated, some scientists believed that a huge
nuclear chain reaction would occur, blowing the earth to pieces. Calculate the energy
that would be required to disassemble the earth completely into pieces totally
separate from each other.

Answer 1

Let's assume that to completely disintegrate something, we have to move all of its pieces completely away from each other. In our case, we'll consider only the gravitationally bound atoms and molecules that constitute the Earth. The atoms within molecules are bound to one another by the electromagnetic force. We want to stick solely to the force of gravity for this analysis, so we'll not consider the ripping apart of molecules.

By symmetry, the energy required to separate all the Earth's bits is equal in magnitude but opposite in sign to the potential energy lost by the Earth's mass as it assembled itself from the protoplanetary disk that once stood where the solar system now exists. The gravitational binding energy (also called gravitational self energy) of a system of particles is the change in the system's potential energy as all its bits fell to the bottom of the system's gravitational potential well.
Or, we can approach it from the other direction: the binding energy is the opposite of the system's change in potential energy as its bits are moved an infinite distance apart. This works as well for gravitationally bound systems like planets as it does for molecules bound by the electromagnetic force and atomic nuclei bound by the strong nuclear force. In the case of the Earth, its gravitational binding energy is equal in magnitude and opposite in sign to the energy required to move its gravitationally bound atoms and molecules an infinite distance apart.

the total change in potential energy as the Earth is disassembled using average density becomes

Using the average density of the Earth = 5.51394