Nitrogen and oxygen both are constituents of our atmosphere and make up about 78% and 21% of our air.
Nitrogen gas has a molecular formula of N2 and a molecular weight of 28 g/mol. Thus, 1 mole of nitrogen gas weighs 28 g. Similarly, oxygen gas has a molecular formula of O2 and a molecular weight of 32 g/mol. Thus, 1 mole of oxygen gas weighs 32 g.
At STP, 1 mole of each gas occupies 22.4 l. Thus, in case of nitrogen, there would be 28 g in 22.4 l of nitrogen gas. Similarly, there would be 32 g oxygen in 22.4 l gas. This means that there is more oxygen than nitrogen in the same volume of gas. Thus, oxygen is heavier than nitrogen gas and not the other way around.
Hope this helps.
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