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If I had $\ce{H2O2}$ in a beaker and I put $\ce{H2}$ and $\ce{O2}$ gas in that beaker, what would happen? $\ce{H2}$ will combine with $\ce{H2O2}$, or a $\ce{O}$ atom in $\ce{H2O2}$ will separate and make bonds with $\ce{H2}$ gas?

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Light can cause the hydrogen peroxide to decompose:

$$\ce{H2O2->2 OH}$$

The $\ce{OH}$ can react with $\ce{H2}$

$$\ce{OH + H2 -> H2O + H}$$

Also, $\ce{OH}$ is an important chemical in atmospheric chemistry. It takes part in several reactions. In the troposphere, it takes part in the following reactions:

$$\ce{OH + CO-> CO2 + H}$$ $$\ce{OH + H2O2-> H2O + HO2}$$ $$\ce{OH + O3-> HO2 + O2}$$ $$\ce{CH4 + OH-> CH3 + O2}$$

The products of these reactions further react with other gases and vapors in the atmosphere. You can learn more about it here.

The behavior of chemicals is complex. "What will happen?" is a question that is almost always too broad.

Source:

Introduction to Atmospheric Chemistry by Daniel J. Jacob, Princeton University Press, 1999.

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