Question

Estimate the mean free path and collision frequency of a nitrogen molecule in a cylinder containing nitrogen at 2.0 atm and temperature 17 °C. Take the radius of a nitrogen molecule to be roughly 1.0 Å. Compare the collision time with the time the molecule moves freely between two successive collisions (Molecular mass of N₂ = 28.0 u).

Answer 1

Mean free path = 1.11 × 10^–7 m

Collision frequency = 4.58 × 10⁹ s^–1

Successive collision time ≈ 500 × (Collision time)

Pressure inside the cylinder containing nitrogen, P = 2.0 atm = 2.026 × 10⁵ Pa

Temperature inside the cylinder, T = 17°C =290 K

Radius of a nitrogen molecule, r = 1.0 Å = 1 × 10¹⁰ m

Diameter, d = 2 × 1 × 10¹⁰ = 2 × 10¹⁰ m

Molecular mass of nitrogen, M = 28.0 g = 28 × 10^–3 kg

The root mean square speed of nitrogen is given by the relation:

Where,

R is the universal gas constant = 8.314 J mole^–1 K^–1

= 508.26 m/s

The mean free path (l) is given by the relation:

Where,

k is the Boltzmann constant = 1.38 × 10^–23 kg m² s^–2K^–1

= 1.11 × 10^–7 m

Collision frequency

= 4.58 × 10⁹ s^–1

Collision time is given as:

= 3.93 × 10^–13 s

Time taken between successive collisions:

= 2.18 × 10^–10 s

∴

Hence, the time taken between successive collisions is 500 times the time taken for a collision.