If n is the number density and d is the diameter of the molecule, then the average distance covered by a molecule between two successive collisions (i.e. mean free path) is represented by :

Solution

Physics sir · 4 steps

Picture a tube of radius d around the molecule's path; the free path is the length holding one molecule.

1. Collision tube. Two molecules of diameter d touch when their centres are d apart. So a moving molecule hits every molecule whose centre lies inside a tube of radius d around its path. This tube has cross-section π d².
2. One hit per path. Over a length λ this tube has volume π d² λ and holds n π d² λ molecules. One collision per free path means n π d² λ = 1, so λ = 1/(n π d²) if the other molecules stood still.
3. Others move too. The other molecules are also moving. Their average relative speed is √2 times the average speed, so collisions happen √2 times as often and λ shrinks by that factor.

   λ = 1/(√2 n π d²)

4. So. With n in m⁻³ and d² in m², the product n π d² is in m⁻¹. So λ comes out in metres, as a length should. The answer is (a).

The idea to keep. Mean free path falls as the gas gets denser and as the molecules get bigger: λ = 1/(√2 n π d²). A quick unit check tells you whether a formula even gives a length.

Answer from the NTA final answer key. Solution by PrepWiser.