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In 1811 Amadeo Avogadro proposed his hypothesis. According to his hypothesis under the similar condition of temperature and pressure equal volume of all gases contain the same number of molecules.

i] To establish the relation between molecular weight and vapour density or,
Prove that: Molecular Weight = 2 × Vapour Density
Mol. Wt. = 2 × V.D.

Vapour density is the ratio of weight of certain volume of a gas to the weight of same volume of hydrogen under the similar condition of temperature and pressure.

ii] Elementary Gases are diatomic gases
a] Hydrogen is a diatomic gas.

n molecules                   n molecules                 2n molecules
1 molecule                    1 molecule                   2 molecules
1/2 molecule                 1/2 molecule                1 molecule

Hydrochloric Acid reacts with base to give only one type of salt. So one molecule of HCl contains only one atom of hydrogen which comes from 1/2 molecule of hydrogen.
Thus,
1/2 molecule of hydrogen = 1 atom of hydrogen
1 molecule of hydrogen = 2 atoms of hydrogen
Hence, Hydrogen is a diatomic gas.

b] Chlorine is a diatomic gas.

n molecules                   n molecules                 2n molecules
1 molecule                    1 molecule                   2 molecules
1/2 molecule                 1/2 molecule                1 molecule

Experimentally it has been found that molecular weight of HCl is 36.5 and it contains one atom of hydrogen. So weight of chlorine in a molecule of hydrogen chloride is 35.5. This weight is atomic weight of chlorine. So one molecule of HCl contains one atom of Chlorine which from 1/2 molecule of Chlorine.

Hence,
1/2 molecule of Chlorine = 1 atom of Chlorine
1 molecule of Chlorine = 2 atoms of Chlorine
Thus, Chlorine is a diatomic gas.

c] Nitrogen is a diatomic gas. n molecules                   n molecules                 2n molecules
1 molecule                    1 molecule                   2 molecules
1/2 molecule                 1/2 molecule                1 molecule

Experimentally it has found that molecular weight of Nitric Oxide is 30, and it contains one atom of oxygen. So weight of nitrogen in a molecule of nitric oxide is 14. This weight is atomic weight of nitrogen. So one molecule of nitric oxide contains one atom of nitrogen which comes from 1/2 molecule of nitrogen.
Hence,
1/2 molecule of nitrogen = 1 atom of nitrogen
1 molecule of nitrogen = 2 atoms of nitrogen
Thus, Nitrogen is a diatomic gas.

d] Oxygen is a diatomic gas: n molecules                   n molecules                 2n molecules
1 molecule                    1 molecule                   2 molecules
1/2 molecule                 1/2 molecule                1 molecule

Experimentally it has found that molecular weight of Nitric Oxide is 30 and it contains one atom of nitrogen. So weight of oxygen in a molecule of nitric acid is 16. This weight is atomic weight of oxygen. So one molecule of nitric oxide contains one atom of oxygen, which comes from 1/2 molecule of oxygen.
Hence,
1/2 molecule of oxygen = 1 atom of oxygen
1 molecule of oxygen    =  2 atoms of oxygen
Thus, oxygen is a diatomic gas.

iii] Molar Volume of each gas at STP is 22.4 litre
According to application of Avogadro's hypothesis
Molecular Weight = 2 × Vapour Density
or,