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The output from an anerobic kitchen waste digester is mostly $\ce{CH4}$, and $\ce{CO2}$. If the methane is collected over water; algae/moss may use the $\ce{CO2}$ to grow. Yet oxygen is released ... which, together with methane, is a potent environment.

Will the oxygen, in this closed jar filled with water+algae+moss, recombine to form $\ce{CO2}$ and $\ce{H2O}$? Is there a means to prevent methane combining with oxygen released by the algae?

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I doubt that the conditions in an Anaerobic digester (organic acids, ammonia (!), etc) allow algae to grow. If they do, the oxygen they set free will create oxic conditions and stop the AD - Methanogens are obligate anaerobe bacteria. In nature, Methanogens and Algae inhabit different habitats - methanogens in the mud at the bottom of a swamp, algae (maybe) in the water above.

Processes for separating $\ce{CO2}$ from $\ce{CH4}$ usually work on the $\ce{CO2}$ (absorption in amines or water under high pressure), so it would be hard to pull the $\ce{CH4}$ from the headspace of your vessel - but given the biological points, this is a non-issue.

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Will the oxygen, in this closed jar filled with water+algae+moss, recombine to form $\ce{CO2}$ and $\ce{H2O}$?

Very slowly, you will not see this. However, it is much worse, as explosive composition of $\ce{CH4}$ and $\ce{O2}$ will form.

In industrial applications it is possible to separate $\ce{CO2}$ from $\ce{CH4}$ and them move $\ce{CO2}$ into separate chamber with algae.

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