Question

Derive the Avogadro's law on the basis of kinetic theory.

Answer 1

Avogadro's hypothesis:-

Consider two samples A and B of the gases each having volume V and pressure p.

Let m₁ and m₂ be the mass of each molecule of gas A and gas B respectively. N₁ and N₂ be the number of molecules of gas A and B respectively.

According to kinetic theory

p = {1mN}/{3V}C²    ....(i)

Therefore, for gas A, p = 1/3 {m₁N₁}/{V} C²₁,

where C₁ = r.m.s. velocity of the molecules of gas A.

And for gas B, p = 1/3 {m₂N₂}/{V}C₂²    ....(ii)

where C₂ = r.m.s. velocity of the molecule of gas B.

From (i) and (ii),

1/3 {m₁N₁}/{V} C₁² = 1/3 {m₂N₂}/{V}C₂²

or, m₁N₁C₁² = m₂N₂C₂²   ....(iii)

If the temperature of gas A and gas B is same, then their average K.E./ molecule is also same.

Thus, 1/2 m₁C₁² = 1/2 m₂C₂²

or, m₁C₁² = m₂C₂²    ...(iv)

Dividing (iii) by (iv), we have

N₁ = N₂   (Avogadro's hyppothesis)

Thus, equal volumes of all gases under similar conditions of temperature and pressure have the same number of molecules.