Calculate the value of mean free path (λ) for oxygen molecules at temperature 27°C and pressure 1.01 × 10⁵ Pa. Assume the molecular diameter 0.3 nm and the gas is ideal. (k = 1.38 × 10⁻²³ JK⁻¹)

Solution

Physics sir · 4 steps

Use λ = kT/(√2 π d² p) and keep every quantity in SI units before you divide.

1. The formula. Mean free path is the average distance a molecule travels between two collisions. For an ideal gas it depends on temperature, pressure and the molecular diameter d. The √2 appears because the other molecules are also moving.

   λ = kT/(√2 π d² p)

2. Numerator. Convert the temperature first: T = 27 + 273 = 300 K. Then kT = 1.38 × 10⁻²³ × 300 = 4.14 × 10⁻²¹ J.
3. Denominator. With d = 0.3 × 10⁻⁹ m, d² = 9 × 10⁻²⁰ m². So √2 π d² p = 4.443 × 9 × 10⁻²⁰ × 1.01 × 10⁵ = 4.04 × 10⁻¹⁴ N.
4. So. λ = (4.14 × 10⁻²¹)/(4.04 × 10⁻¹⁴) = 1.02 × 10⁻⁷ m ≈ 102 nm, which is option (d).

The idea to keep. Mean free path grows with temperature and falls with pressure and with d². Always use kelvin and metres, and do not forget the √2 in the denominator.

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