A system consists of two types of gas molecules A and B having same number density 2 × 10²⁵/m³. The diameter of A and B are 10 Aᵒ and 5 Aᵒ respectively. They suffer collision at room temperature. The ratio of average distance covered by the molecule A to that of B between two successive collision is ____________ × 10⁻²

Solution

Physics sir · 4 steps

Same number density, so the mean free path depends only on the square of the diameter.

1. Mean free path. The average distance a molecule travels between two successive collisions is its mean free path. A fatter molecule sweeps out a bigger collision cross-section π d², so it hits others sooner.

   λ = 1/(√2 π d² n)

2. Take the ratio. Both gases have n = 2 × 10²⁵ m⁻³, so n, √2 and π cancel. That leaves λ_A/λ_B = d_B²/d_A².
3. Put in numbers. λ_A/λ_B = (5)²/(10)² = 25/100 = 0.25. The ångström units cancel, so you do not need to convert them.
4. So. λ_A/λ_B = 0.25 = 25 × 10⁻², so the answer is 25.

The common slip: 50. Taking λ proportional to 1/d instead of 1/d² gives a ratio of 0.5, which is 50, not 25.

The idea to keep. Mean free path goes as 1/(d² n). Double the diameter and the mean free path drops to a quarter, because the collision cross-section is an area.

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