Thermal Physics - Mean Free Path

The mean free path for the molecules of a gas is approximately given by 1/ησ, where η is the number density and σ is the collision cross section. The mean free path for air molecules at room conditions is approximately

A. 10⁻⁴ m
B. 10⁻⁷ m
C. 10⁻¹⁰ m
D. 10⁻¹³ m
E. 10⁻¹⁶ m

(GR9677 #16)

Solution:

Mean free path = 1/ησ

Number density, η = N/V  
Cross section area, σ = πr² 

For Ideal Gas: PV = NkT

1/ησ = V / Nπr²  = NkT / PNπr² = kT / Pπr²   

k = 1.38 × 10⁻²³ Joule/K
Radius of atom is in order of Angstrom: 10⁻¹⁰ m
P (STP) = 1 atm = 10⁵ Newton/meter²
T (STP) = 0 °C = 32 °F = 273.15 K ≈ 2 × 10² K

kT / Pπr² = (1.38 × 10⁻²³ × 2 ×10²) / (π ×10⁵ × 10⁻²⁰) 
= (1.38 × 2 / π) 10^{−23+2−5+20} ≈ 10⁻⁷

Answer: B