The number of air molecules per cm³ increased from 3×10¹⁹ to 12×10¹⁹. The ratio of collision frequency of air molecules before and after the increase in number respectively is:

Solution

Physics sir · 3 steps

Speed stays the same, so collision frequency just follows the number density: 3 to 12 means one quarter.

1. Collision frequency. A molecule collides about v̄/λ times per second. Here v̄ is its mean speed and λ is the mean free path.

   f = v̄/λ = √2 π d² n v̄ since λ = 1/(√2 π d² n)

2. What changes. The molecular diameter d is fixed. The mean speed v̄ depends only on temperature, which is unchanged. So only n changes, and f ∝ n.
3. So. f₁/f₂ = n₁/n₂ = (3 × 10¹⁹)/(12 × 10¹⁹) = 0.25, which is option (b).

The idea to keep. Mean free path goes as 1/n, and collision frequency goes as n when temperature is fixed. Pack in more molecules and each one gets hit more often, in direct proportion.

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