Question

Prove the following:
(a) Oxygen is a diatomic molecule.
(b) 2 × V.D. = Molecular mass.
(c) One mole of any gas contains the same number of molecules.

Answer 1

(a) Let us consider the reaction between carbon and oxygen to form carbon dioxide.
Carbon + Oxygen $\to$ CO₂
1 mole 1 mole 1 mole [according to Avogadro's Law]

1 molecule 1 molecule 1 molecule
As carbon dioxide (CO₂) contains two oxygen atoms, the number of oxygen atoms on reactant side must be two. But reaction uses only one molecule of oxygen to produce one molecule of carbon dioxide. Then, oxygen molecule must contain two atoms of oxygen, i.e., oxygen molecule is diatomic.

(b) Vapour density is the ratio of same volume of mass of given gas to the mass of hydrogen gas at fixed temperature and pressure conditions.
Vapour density of a gas = $\frac{\mathrm{Mass}\mathrm{of}\mathrm{gas}\mathrm{at}\mathrm{volume}V}{\mathrm{Mass}\mathrm{of}\mathrm{hydrogen}\mathrm{gas}\mathrm{at}\mathrm{volume}V}$
Now, according to Avogadro's law at fixed temperature and pressure, volumes can be substituted by molecules.
Vapour density of a gas = $\frac{\mathrm{Molecular}\mathrm{mass}\mathrm{of}\mathrm{gas}}{\mathrm{Molecular}\mathrm{mass}\mathrm{of}\mathrm{hydrogen}\mathrm{gas}}$

Vapour density of a gas = $\frac{\mathrm{Molecular}\mathrm{mass}\mathrm{of}\mathrm{gas}}{2}$
Vapour density $\times$ 2 = Molecular mass

(c) Mass of one mole of a gas = Number of molecules in one mole of gas $\times$ Mass of one molecule of gas
Let the mass of one mole of gas = x grams
∴​ Mass of one molecule of the same gas = ​x a.m.u.
So, the equation will be as follows:
x grams of gas = ​Number of molecules in one mole of gas $\times$ x a.m.u.
Number of molecules in one mole of gas = $\frac{1\mathrm{gram}\mathrm{of}\mathrm{gas}}{1\mathrm{amu}\mathrm{of}\mathrm{gas}}$ .....(i)
1 a.m.u. = 1.66 $\times$ 10⁻²⁴ g
Substituting this value in relation to equation (i), we get:
Number of molecules in one mole of gas = $\frac{1\mathrm{gram}\mathrm{of}\mathrm{gas}}{1.66\times {10}^{-24}\mathrm{grams}\mathrm{of}\mathrm{gas}}$ = 6.023 $\times$10²³
Hence, one mole of any gas contains the same number of molecules, which is equal to 6.023 $\times$10²³.