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In the reduction of copper oxide by hydrogen in the lab to copper, why does hydrogen need to flow through the tube even after the reduction is completed and the burner is turned off until the tube is cool?

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As the system cools, the gases will contract. Unless a positive flow of hydrogen (or some other inert gas) is maintained during the cooling process, this will result in room air being pulled into your tube. The oxygen present in air would then oxidize some of $\ce{Cu}$ back to $\ce{CuO}$, so long as some regions within the tube remained hot enough for this reaction to take place (at least $\pu{300^oC}$ or so).

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