Mean Free Path Calculator

Created by Dominik Czernia, PhD candidate
Last updated: Dec 06, 2021

This mean free path calculator will help you calculate the mean free path of a particle with the specific diameter in an ideal gas. An ideal gas consists of a large number of particles (atoms or molecules) which are in constant rapid motion and can collide with each other. During these perfectly elastic collisions, particles can change their directions or energies. The mean free path is the average distance that a particle travels in a material medium, between successive collisions with particles of that medium. The term "mean free path" is used in many areas of physics.

Do you want to learn how to calculate the mean free path? In the following text, you will find a more detailed mean free path definition and the mean free path formula. Use our calculator to estimate the average distance between particles' collisions of every ideal gas you want. All you need is to provide the ideal gas parameters (pressure and temperature) and an effective particle diameter to the mean free path equation. At the end of this text, you can find a table with the atomic radii of some elements.

If you want to analyze properties of real gases, don't hesitate and check out our van der Waals equation calculator!

Mean free path definition

In gases, particles have full freedom of movement - there are no chemical bonds between them. All of them are in constant motion in the space occupied by the gas and never stop in one place. There are virtually no long-range interactions between molecules, and the existing ones are very weak. The only way in which particles can interact are collisions between them. Moreover, if the gas is placed in a container, its particles constantly collide with the walls of the container, exerting specific and constant pressure on them. Have you ever wondered how fast do particles move in gases? Check it out using our particles velocity calculator!

Red and blue particles moving in a box animation

Look at the above animation and focus on one specific particle. How far does it travel without any collision? The answer will always be different, because of the chaotic motion of those particles. Nevertheless, we can calculate what is the average distance traveled by a moving particle. We call it the mean free path and it is often used in:

  • radiography to measure the thickness of a material,
  • particle physics to measure the magnitude of the interactions of the high-energy photons with a material (radiation length),
  • electronics to describe the electrical mobility,
  • diffusion and transport phenomena, for example to determine the Knudsen number which helps to decide whether we can use statistical mechanics or the continuum mechanics,
  • also nuclear physics, optics, astronomy or acoustics.

How to calculate mean free path? - mean free path equation

You can easily calculate the mean free path of particles of a gas, provided that this gas can be described with the ideal gas law equation. Our mean free path calculator uses the below mean free path equation:

λ = k * T / (√2 * π * d² * p)

where

  • λ is the mean free path expressed in the length units,
  • T is the temperature of the gas,
  • p is the pressure of the gas,
  • d is the kinetic diameter of a particle,
  • k is the Boltzmann constant k = 1.380649 * 10^(−23) J / K.

In fact, the diameter of a particle is not well defined, because gas particles do not behave like hard spheres. We use only the effective diameter of the atom which is the minimum distance at which two particles of the same gas are approaching each other during a collision. In other words, it is the distance between the centers of these particles at the moment of collision.

The mean free path of gas molecules is often used to measure the quality of the vacuum. For example, the mean free path of air particles at room temperature and atmospheric pressure 1013 hPa equals 68 nm. However, if we decrease pressure to high vacuum ~10^(-5) Pa, the mean free path is approximately 1 km.

Mean free path formula - kinetic radii table

Our mean free path formula requires kinetic diameter to perform proper calculations. You can find both radii and diameters (two times radius) of the molecules of some gases in the below table (expressed in picometers - pm) which we created based on the on Wikipedia.

Molecule Atomic radius (pm) Atomic diameter (pm)
Hydrogen (H₂) 144.5 289
Helium (He) 130 260
Nitrogen (N₂) 182 364
Oxygen (O₂) 173 346
Neon (Ne) 137.5 275
Chlorine (Cl) 160 320
Argon (Ar) 170 340
Bromine (Br) 175 350
Krypton (Kr) 180 360
Xenon (Xe) 198 396
Water (H₂O) 132.5 265
Carbon monoxide (CO) 188 376
Carbon dioxide (CO₂) 165 330
Nitric oxide (NO) 158.5 317
Nitrous oxide (N₂O) 165 330
Sulfur dioxide (SO₂) 180 360
Hydrogen sulfide (H₂S) 180 360
Hydrogen chloride (HCl) 160 320
Hydrogen bromide (HBr) 175 350
Ammonia (NH₃) 130 260
Sulfur hexafluoride (SF₆) 275 550
Carbon tetrachloride (CCl₄) 295 590
Methane (CH₄) 190 380
Acetylene (C₂H₂) 165 330
Ethylene (C₂H₄) 195 390
Propylene (C₃H₆) 225 450
Propane (C₃H₈) 215 430
Benzene (C₆H₆) 292.5 585
Dominik Czernia, PhD candidate
Particles
Acetylene (C₂H₂)
Particle diameter
pm
Pressure
atm
Temperature
K
Mean free path
nm
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