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Beryllium carbonate is reported to be unstable under normal conditions and should only be kept in a atmosphere of $\ce{CO2}$. Why and how this stability is maintained? Why does this compound decompose when kept in an atmosphere of oxygen?

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Why and how is this stability maintained?

Beryllium carbonate decomposes into Beryllium oxide and Carbon dioxide. $$\ce{BeCO3 <=> BeO + CO2}$$

According to Le Chatelier's principle one simply has to store it under a $\ce{CO2}$ atmosphere to push the equilibrium to the left. $$\ce{BeCO3 <<=> BeO + CO2}~\text{in excess}$$

Why this compound decomposes when kept in atmosphere of oxygen?

When there is no $\ce{CO2}$ atmosphere the equilibrium is pushed to the right, which would be the "normal" case under standard conditions. $$\ce{BeCO3 <=>> BeO + CO2}$$

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