Mean Free Path Calculator Formula

Understand the math behind the mean free path calculator. Each variable explained with a worked example.

Formulas Used

Mean Free Path

mean_free_path = (1.38e-23 * temp) / (1.4142 * pi * pow(diameter_m, 2) * pressure_pa)

Variables

Variable	Description	Default
`temp`	Temperature (T)(K)	298
`pressure_pa`	Pressure(Pa)	101325
`diameter_m`	Molecular Diameter(m)	3.7e-10

How It Works

Mean Free Path

Formula

λ = kT / (√2 × π × d² × P)

Where:

λ = mean free path (m)

k = Boltzmann constant = 1.38 × 10⁻²³ J/K

T = temperature (K)

d = molecular diameter (m)

P = pressure (Pa)

The mean free path increases with temperature and decreases with pressure and molecular size.

Worked Example

N₂ molecules (d = 3.7 × 10⁻¹⁰ m) at 298 K and 101325 Pa.

temp = 298pressure_pa = 101325diameter_m = 3.7e-10

01λ = kT / (√2 × π × d² × P)
02λ = (1.38×10⁻²³ × 298) / (1.414 × 3.14159 × (3.7×10⁻¹⁰)² × 101325)
03λ = 4.11×10⁻²¹ / (1.414 × 3.14159 × 1.369×10⁻¹⁹ × 101325)
04λ ≈ 6.6 × 10⁻⁸ m = 66 nm

Frequently Asked Questions

What is the mean free path?

It is the average distance a gas molecule travels between successive collisions with other molecules. At atmospheric pressure, it is typically tens of nanometers.

How does pressure affect mean free path?

Lower pressure means fewer molecules per volume, so molecules travel farther between collisions. In vacuum chambers, mean free paths can be meters.

Why does molecular diameter matter?

Larger molecules have bigger collision cross-sections, making collisions more frequent and the mean free path shorter.

Ready to run the numbers?

Open Mean Free Path Calculator