Discover a fundamental quantity of physics with the mean free path calculator. In this short article, you will learn what the mean free path is and why physicists need it to model the behavior of gases. We will introduce you to the most famous equation for the mean free path, the one we implemented in our tool, and give you some handy examples: from our atmosphere to the mean free path in the vacuum between stars.

What is the mean free path?

The mean free path is a quantity widely used in physics that describes a statistical measure of the distance/time a particle can travel before a collision that changes its motion in a substantial manner.

Thanks to the particle-wave dualism (we talked about this basic concept in quantum mechanics at our De Broglie wavelength calculator), we can define the mean free path both for many different objects:

  • In gases, the mean free path is an important measure, correlated mainly to the pressure in a container. The mean free path is a fundamental quantity in experimental physics, particularly in high vacuum applications.
  • We can define the mean free path for photons. In this case, the mean free path equation describes the interaction between light and a material's atoms: the applications are primarily in radiography.
  • In electronics, we calculate the mean free path for electrons in a conductor: this quantity is often related to the quality of the material itself.

There are many other applications for the mean free path, but the one regarding gases is the most intuitive and widespread. In the next section, we will learn how to calculate the mean free path.

How to calculate the mean free path: formula for a gas

The formula for the mean free path of a gas is derived for ideal gases: we are talking of low density gases, where intermolecular forces are negligible. Ideal gases are one of the fundamental concepts of thermodynamics since many models describing complex systems have their foundations there: we talked in depth about them in many tools: the ideal gas law calculator, the combined gas law calculator, and many more!

The equation for the mean free path in a gas is:

λ=kT2πd2p\lambda = \frac{k\cdot T }{\sqrt{2\cdot\pi\cdot d^2 \cdot p}}

Where:

  • λ\lambda — The mean free path of the gas molecules,
  • TT — The temperature of the gas
  • PP — The pressure;
  • dd — The kinetic diameter of the molecule;
  • kB=1.380649×1023k_{\text{B}} = 1.380649 \times 10^{−23} — The Boltzmann constant.

The measurement unit of the mean free path is, of course, a length one. In the scientific environment, you are likely to find the meter, but don't be scared of seeing other units like the kilometer. Why?

Because of how gases behave at really low temperatures or really high pressure. The two concepts are, on the surface, unrelated; however, they share essential consequences when we calculate the mean free path.
For a lower temperature, the kinetic energy of the gas molecules decreases: the particles quite literally slow down (they would stop entirely at the absolute zero). For extremely low pressures, the number of molecules is so small that the probability of two of them meeting is ridiculously low. Both factors affect the number of collisions.

Regarding collisions, we need to dedicate a few words to an essential quantity in the equation for the mean free path: the kinetic diameter.

The kinetic diameter in the mean free path formula

A collision between two human-sized projectiles happens if the two objects physically touch each other. For small particles lying halfway on the atomic scale, talking of a collision require a bit of statistics and more thought. In the mean free path formula, we count the effective diameter of the gas molecule that would involve a collision. This quantity is larger than the actual diameter of the molecule (and of the electron clouds of its composing atoms): we calculate it with the equation:

d2=12πλnd^2 = \frac{1}{\sqrt{2}\cdot \pi\cdot \lambda\cdot n}

nn is the number density of the molecules, and λ\lambda is the mean free path: the quantities are deeply interconnected. Notice how the quantity is already squared.

You can find tabulated values for the kinetic diameter online for many gases and in various conditions.

The result of the mean free path equation: how long before a collision?

At room temperature and atmospheric pressure, a nitrogen molecule has a mean free path just below 70 nm70\ \text{nm}. Since the average speed of a molecule in air is about 500 m/s500\ \text{m}/\text{s}, the time between collisions is 0.14 ns0.14\ \text{ns}. It's needless to say: this is a short amount of time.

LHC (Large Hadron Collider), the particles accelerator in Geneva, operate at the amazing conditions of T=2 KT = 2\ \text{K} and P=1010 mbarP=10^{-10}\ \text{mbar}. Under these parameters, a Helium molecule (the most likely found in those chambers) is so lonely that we need to change the units of the mean free path: from nanometers, we switch to kilometers: the distance required to collide, on average, with another helium molecules is about $$10\ \text{km}**. Play with our mean free path calculator to discover other surprising aspects of the physics of the extremes.

For example, if you are asking "what is the mean free path in the vacuum", we can give you the value of the pressure in interstellar space, 4.0×1022 atm4.0\times10^{-22}\ \text{atm}, the temperature (again 2 K2\ \text{K}). Input these values in our tool: the mean free path is a staggering 1,836,013,911 km1,836,013,911\ \text{km}. This result, however, doesn't reflect the true nature of the vacuum: more accurate models that consider the presence of fields and other effects negligible in our daily experiences are required.

Davide Borchia
Particles
Acetylene (C₂H₂)
Particle diameter
pm
Pressure
atm
Temperature
K
Mean free path
nm
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