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Assume you have burned some substance A in an airtight container until the fire goes out due to insufficient oxygen. You then place some burning substance B in the container through an airlock. Are there substances A and B such that the substance B would continue to burn for a significant amount of time using the oxygen left behind after substance A burned out?

In other words, do all materials require the same concentration of oxygen to sustain a flame?

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No, different substances will cease combustion at different concentrations of oxygen. The concentration needed can be described by the limiting oxygen index. This is the reason that the use of pure compressed oxygen can be so hazardous; not only will materials burn that would not have burned before, but now materials that do burn require a lower temperature/energy input to ignite.

You can understand this conceptually when you consider where the heat goes. On combustion in air some of the heat must be used to heat the $\pu{79\!\%}$ of the air that isn't oxygen and that heat is wasted in the inert gas and the heat is diluted. That means only ~$\pu{20\!\%}$ of the heat goes into activation energy to propogate the combustion. When the oxygen is more concentrated, less heat goes into inert gases and more goes into oxygen to propagate the combustion and vise-versa.

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Since A.K. has already answered the question you're really asking, I feel it's reasonable to interpret your question literally and answer, "No, because it's possible to have flames with no oxygen at all."

The flames in this Periodic Videos video are from iron burning in fluorine. The experiment is done in the open air (wot no fume cupboard?) so the later glowing and sparking is due to the hot iron reacting with atmospheric oxygen. Later, the video also shows charcoal, cotton wool, sulphur and even iodine burning in fluorine. Again, some of those (especially the cotton wool) get hot enough that they continue to burn in atmospheric oxygen after the fluorine has been turned off, but the initial flames are from the reaction with fluorine.

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    $\begingroup$ Indeed, there are reactive metals that can burn in much less reactive gases, such as carbon dioxide, nitrogen and water vapour. Hence if material A is a hydrocarbon, and consumes (nearly) all the oxygen in Steve B's vessel, and material B is, say, lithium, then material B can burn in the products of combustion! $\endgroup$ – Securiger Sep 2 '18 at 12:46

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