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If the maganese dioxide is acting as a catalyst then the minimum amount would be one molecule. As a catalyst is not consumed in a chemical reaction (or replaces itself with an exact dulipcate of itself so the end result is the same.) Then that single molecule will work for eternity converting reactants into product. The question needs to specify reaction rates and expected yields before we can get into particulars.

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Q: Determination of the minimum quantity of maganese dioxide required as a catalyst for the preparation of oxygen gas?
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