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Before the first nuclear weapon was dropped, a scientist (whose name has escaped me) had a thought: could it ignite the atmosphere? But this was during World War II, when they didn't have the maths it would take to determine whether they could have ignited the atmosphere, so they dropped it anyway.
How would one ignite the atmosphere? How plausible is it, and how easy would it be for me to do this?
It was something that was brought up, but it was fairly quickly debunked. The concern was not about a chemical reaction (the atmosphere is pretty chemically inert as its two main constituents don't tend to react with each other). Rather they were worried that a nuclear chain reaction of either nitrogen-nitrogen fusion or nitrogen-hydrogen fusion (with the hydrogen being supplied by boiling the oceans and disassociating the water). Here's a contemporary paper about it, showing that it won't happen: http://www.fas.org/sgp/othergov/doe/lanl/docs1/00329010.pdf
Roughly speaking, they show that at any temperature the heat radiated away is greater than the heat added by nitrogen-nitrogen fusion reactions, given the density of nitrogen in the atmosphere and assuming that any physical collision of sufficient energy results in fusion (because they did not know the true reaction cross section). This means that energy is always lost so the reaction cannot be self sustaining. They also showed that N-H physical collisions are less likely and that other processes they haven't accounted for take more energy out, so overall they show that it doesn't happen even taking generous assumptions in favour of it happening.
So I'm afraid the answer is you can't under the real laws of physics with an earth like atmospheric composition. If you can put large quantities of something flammable or fusible/fissile into the atmosphere then it becomes much easier, obviously. If we're not talking about the real world and you want to change the laws of physics in a way that isn't too noticeable, then perhaps something like increasing the size of atomic nuclei so fusion interactions become more likely would make the situation the scientists feared possible (ignoring the knock on effects in stars etc).
The question has the facts slightly confused. It was Edward Teller, always an enthusiast for very large explosions, who raised the question. This wasn't a matter of ordinary combustion, because trying to burn nitrogen with oxygen doesn't release heat, but absorbs it. That's easy to determine: the atmosphere has not gone up in flames due to lightning strikes in the 850 million years since it started to contain significant amount of oxygen.
The actual fear was of a nuclear chain reaction, fusing the nuclei of nitrogen atoms. If this could be started and become self sustaining, then the entire atmosphere would go up in a nuclear explosion. This would certainly wipe out all life on Earth. However, this is physically impossible.
The mathematics and physics to calculate this was available at the time and Hans Bethe and Emil Konopinski did the calculations and showed that it was impossible; with a large margin of error. A few years later, Konopinski, C. Marvin and Teller did the calculation for thermonuclear weapons, and showed that it was impossible for them, too. In fact, their report is available on-line: here.
To summarize, the heat produced by such fusion is radiated away much too fast for the reaction to be self-sustaining. You probably will get some nitrogen fusion within a nuclear explosion, but the reaction stops very quickly. Practical fusion explosions need their fuel to be at vastly greater density than the atmosphere, and to be confined in some way (such as in the core of a star). These conditions just aren't available for a bomb set off in the atmosphere.
So, this was not a concern at the time of the first nuclear test (which was not dropped, but set off on top of a tower). Enrico Fermi was offering to take bets on its likelihood, but this was a joke.
The idea of igniting the atmosphere was that they were concerned that the tremendously focused heat of that nuclear explosion could create a chain reaction. Gases combust at a higher rate at high temperatures, and we didn't have very much evidence as to what would happen to gasses at those temperatures. We only had our theoretical models (and some small scale tests). There was concern that at those temperatures, gases in the atmosphere could combust fast enough and hot enough to cause nearby atmosphere to combust too.
Of course, this was put to rest before they tested the bomb. Well, as much as a theory is ever put to rest in science. This is the challenge one faces with new physics. Nobody had experience with a nuclear weapon that big. In science, we constantly have to worry about doing theoretical modeling in a region where that model breaks down due to some unforeseen change. A conscientious scientist worries about this every time they push the boundaries of science.
With modern knowledge, its pretty clear we can't ignite the atmosphere. Not enough oxygen, not enough fuel. It's pretty inert stuff.
In theory one could try to unleash a planetary scale volume of hydrogen and oxygen into the atmosphere to richen the mixture to where it would ignite. However, when you ask "how easy would it be for me to do this" the answer is "it would be a gargantuan feat for all of humanity at best."
Well to be fair, a nuclear explosion is not that hot after all. 10 Million degrees surely sounds like alot but this is "only" 10^7, while the theorethical maximum is somewhere at 10^32.
So, if the source is hot enough, the heat radiation might not be fast enough before the Athmosphere, or most of it has been nitrogen fused.
