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If you put a paper inside an oven vacuum packed and go beyond $250~\mathrm{^\circ C}$, will it burn? There's no oxidizer. If not, what will happen?

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No, the paper will not burn without oxygen being present.

Paper is made primarily of cellulose which is a polymer of glucose.

enter image description here

If you heat paper in a vacuum the cellulose simply decomposes to $\ce{H2O}$, $\ce{CO2}$, $\ce{CO}$ and carbon. As the paper decomposes it will "char" or turn brown to black as the cellulose polymer degrades. Here is a link to an abstract describing the process in more detail.

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    $\begingroup$ Depending on the precise temperature and duration of heating, a lot of interesting degradation products can form before things wind up as water, CO2, CO, or C. For example pyrolysis oil is made by heating biomass -- which like paper contains a lot of cellulose. $\endgroup$ – Curt F. Mar 18 '15 at 3:41
  • $\begingroup$ In addition to @CurtF. comment about yielding liquids you may use, do not forget the wood gas generators especially popular during WWII (e.g., en.wikipedia.org/wiki/Alternative_fuel_vehicle#Wood_gas), and more in detail -- including post WWII examples, en.wikipedia.org/wiki/Wood_gas_generator. Albeit feed by wood, I speculate paper could in principle undergo the same as wood (albeit with less efficiency than wood). $\endgroup$ – Buttonwood Apr 5 at 19:16
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It will decompose into other compounds. This is how charcloth is made. It is called dry distillation or pyrolysis.

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The title poses, "Can paper burn without oxygen or air?" A little sneaky pedantry here from an old-school chemist - No Oxygen present, no air either, nor any mention of a vacuum. This allows us the presence of other strong oxidisers such as Chlorine and other halogens which will support combustion, which is what most people mean by "burn". As an aside, I well remember a chemistry practical demonstration where my teacher plunged a spoon of Phosphorus into a cylinder of pure chlorine to see a pale green flame spontaneusly appear, accompanied by copious white smoke (Phoshorus pentachloride) being emitted. So the answer is yes, but only in very uncommon enviroments, but which are found in the chemical industry where the Fire Service need to be aware of potential fire hazards.

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If you put a paper inside an oven vacuum packed and go beyond $250~\mathrm{^\circ C}$, will it burn? There's no oxidizer. If not, what will happen?

Probably not.

As other answers point out. However, paper is a complex and varied material and some varieties called paper could contain trace quantities of materials that upon heating, could produce oxidizing species. So a hard, firm "No" may not be true in all cases.

Can paper burn without oxygen or air?

Yes!

You can watch iron, sulfur, coal, and cotton spontaneously ignite when exposed to fluorine gas in the video Fluorine - Periodic Table of Videos

I'd say cotton would be a reasonable proxy for paper here. Not only do these materials burn in fluorine, combustion is spontaneous!

enter image description here

Incidentally you can look up individual elements on Periodic Table of Videos at http://www.periodicvideos.com/

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Interestingly, this source on paper bleaching cites a so-called Chloramine-T process, and to quote:

It is a little surprising that an oxidizing agent is stable enough to continue bleaching over an extended period of time...

So, I would speculate that thermal heating of paper containing some moisture which was progressed by a Chloramine-T process is likely to undergo, at least to a limited extent, also some chlorine-based oxidation (which normally occurs slowly over time) during its thermal decomposition.

Suggested chloramine chemistry:

$\ce{NH2Cl + H2O <=> NH3 + HOCl}$

where the hypochlorous acid is a strong bleaching/oxidizing agent and on further heating also:

$\ce{2 HOCl <=> Cl2O + H2O}$

with the liberation of a dichlorine oxide (also a strong oxidizing agent and a source gas for combustion).

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