Very ELI5 explanation.
Every atom in existence wants to be iron. Iron is a perfect stable element. (Another ELI5 topic)
If it is lighter than iron, than by fusing elements together you release energy.
If it is heavier than iron, than by breaking it up, you release energy,
When atomic bomb explodes, you use fission (breaking up) of heavy elements such as Uranium and Plutonium, to release large amount of energy.
The fear and hypothesis was, that this release might be so large, it might actually push surrounding air molecules to undergo fusion (joining into heavier elements) releasing more energy, causing more air atoms to fuse and on and on.
This hypothesis has been mostly dismissed because, the energy dissipation is just too fast and the the air is not dense enough or populated by right elements to sustain this reaction.
The way I understand it, causing a chain fusion reaction in an atmosphere cannot really happen. The one exception being, that if you have an explosion large enough to be able to fuse most of the atmosphere on it’s own, than the energy output from fusing the large portions of the atmosphere might give the explosion sufficient kick to fuse the rest. But at that point, there will be not much left of Earth left anyway so …
Less ELI5 but still ELI<12 Reality is, that pushing atoms into fusion is just so damn hard, and the energy of the blast dissipates very quickly. The H-Bomb that is specifically designed to fuse the most fusible material known to man, packs it around the core of nuclear bomb, and still, it is not 100% efficient (Not surprisingly not much information is publicly available, but I have seen estimations between 25%-60%). Showing exactly how hard it is to achieve fusion reaction by design.
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