In physics and kinetic theory, the mean free path of a particle, such as a molecule, is the average distance the particle travels between collisions with other particles.

The formula for calculating the magnitude of the mean free path depends on the characteristics of the system the particle is in. For a particle with a high velocity relative to the velocities of an ensemble of identical particles with random locations, the following relationship applies:

$\backslash ell\; =\; (n\backslash sigma)^\{-1\},$

Where $\backslash ell$ is the mean free path, n is the number of particles per unit volume, and σ is the effective cross-sectional area for collision. If, on the other hand, the velocities of the identical particles have a Maxwell distribution of velocities, the following relationship applies:

$\backslash ell\; =\; (\backslash sqrt\{2\}\backslash ,\; n\backslash sigma)^\{-1\}.\backslash ,$

Derivation

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Imagine a beam of particles being shot through a target, and consider an infinitesimally thin slab of the target (Figure 1). The atoms that might stop a beam particle are shown in red. The area of the slab is $L^\{2\}$ and its volume is $L^\{2\}dx$. The typical number of stopping atoms in the slab is the concentration $n$ times the volume, i.e., $n\; L^\{2\}dx$. The probability that a beam particle will be stopped in that slab is the net area of the stopping atoms divided by the total area of the slab

$$

P(\mathrm{stopping \ within\ dx}) = \frac{\mathrm{Area_{atoms}}}{\mathrm{Area_{slab}}} = \frac{\sigma n L^{2} dx}{L^{2}} = n \sigma dx

where $\backslash sigma$ is the area (or, more formally, the "scattering cross-section") of one atom.

The drop in beam intensity equals the incoming beam intensity multiplied by the probability of being stopped within the slab

$$

dI = -I n \sigma dx

This is an ordinary differential equation

$$

\frac{dI}{dx} = -I n \sigma \equiv -\frac{I}{\ell}

whose solution is $I\; =\; I\_\{0\}\; e^\{-x/\backslash ell\}$, where $x$ is the distance traveled by the beam through the target and $I\_\{0\}$ is the beam intensity before it entered the target.

$\backslash ell$ is called the mean free path because it equals the mean distance traveled by a beam particle before being stopped

$$

\ell \equiv \frac{1}{n\sigma} = \langle x \rangle \equiv \int dx \ e^{-n \sigma x}

Examples

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A classic application of mean free path is to estimate the size of atoms or molecules. Another important application is in estimating the resistivity of a material from the mean free path of its electrons.

For example, for sound waves in an enclosure, the mean free path is the average distance the wave travels between reflections off the enclosure's walls.

References

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External links

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• Gas Dynamics Toolbox Calculate mean free path for mixtures of gases using VHS model

Statistical mechanics

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