MEAN FREE PATH

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:
:$ell\; =\; (nsigma)^\{-1\},$
Where $ell$ 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:
:$ell\; =\; (sqrt\{2\},\; nsigma)^\{-1\}.,$
Following table lists some typical values for different pressures.

Vacuum range	Pressure in hPa	Molecules / cm3	mean free path
Ambient pressure	1013	2.7 ★ 1019..	68 nm
Low vacuum	300..1	1019..1016	0.1..100 μm
Medium vacuum	1..10-3	1016..1013	0.1..100 mm
High vacuum	10-3..10-7	1013..109	10 cm..1 km
Ultra high vacuum	10-7..10-12	109..104	1 km..105 km
Extremely high vacuum	<10-12	<104	>105 km

Contents

Derivation

Examples

See also

External links

Derivation

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}) =
rac{mathrm{Area_{atoms}}}{mathrm{Area_{slab}}} =
rac{sigma n L^{2} dx}{L^{2}} = n sigma dx

where $sigma$ 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
:$$
rac{dI}{dx} = -I n sigma stackrel{mathrm{def}}{=} -rac{I}{ell}

whose solution is $I\; =\; I\_\{0\}\; e^\{-x/ell\}$,
where $x$ is the distance traveled by the
beam through the target and $I\_\{0\}$ is the beam
intensity before it entered the target.
$ell$ is called the mean free path because
it equals the mean distance traveled by a beam particle
before being stopped. To see this, note that the probability that the a particle is absorbed between x and x+dx is given by
:
Thus the expectation value (or average, or simply mean) of x is
:$$
langle x
angle stackrel{mathrm{def}}{=} int_0^infty x dP(x) = int_0^infty rac{x}{ell} e^{-x/ell} dx = ell

Examples

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.

See also

★ Vacuum

External links

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