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It is known that pure heavy water will quickly get contaminated by light hydrogen atoms if exposed to air due to isotope exchange.

Does the same thing happens when heavy water is stored in plastic container, which also contains light hydrogen atoms?

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If you mix D$_2$O and H$_2$O you quickly get DHO due to the Grotthuss mechanism. I assume this is what you mean by heavy water being contaminated on exposure to humid air. Obviously it wouldn't be contaminated by exposure to dry air because there's no hydrogen present in dry air.

The hydrogen atoms in polyethylene aren't mobile and won't react with D$_2$O. However polyethylene is more porous than you might think and will eventually let water through.

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