Editing Mean free path (section)

==Kinetic theory of gases==
In the [[kinetic theory of gases]], the ''mean free path'' of a particle, such as a [[molecule]], is the average distance the particle travels between collisions with other moving particles. The derivation above assumed the target particles to be at rest; therefore, in reality, the formula <math>\ell = (n\sigma)^{-1}</math> holds for a beam particle with a high speed <math>v</math> relative to the velocities of an ensemble of identical particles with random locations. In that case, the motions of target particles are comparatively negligible, hence the relative velocity <math>v_{\rm rel} \approx v</math>.

If, on the other hand, the beam particle is part of an established equilibrium with identical particles, then the square of relative velocity is:

<math>\langle\mathbf{v}_{\rm relative}^2\rangle =\langle(\mathbf{v}_1-\mathbf{v}_2)^2\rangle =\langle\mathbf{v}_1^2+\mathbf{v}_2^2-2\mathbf{v}_1 \cdot \mathbf{v}_2\rangle.</math>

In equilibrium, <math>\mathbf{v}_1</math> and <math>\mathbf{v}_2</math> are random and uncorrelated, therefore <math>\langle\mathbf{v}_1 \cdot \mathbf{v}_2\rangle =0</math>, and the relative speed is

<math>v_{\rm rel}=\sqrt{\langle\mathbf{v}_{\rm relative}^2 \rangle}
=\sqrt{\langle\mathbf{v}_1^2+\mathbf{v}_2^2\rangle}
=\sqrt{2}v.</math>

This means that the number of collisions is <math>\sqrt{2}</math> times the number with stationary targets. Therefore, the following relationship applies:<ref>S. Chapman and T. G. Cowling, [https://books.google.com/books?id=Cbp5JP2OTrwC&pg=PA88 ''The mathematical theory of non-uniform gases''], 3rd. edition, Cambridge University Press, 1990, {{ISBN|0-521-40844-X}}, p. 88.</ref>

:<math>\ell = (\sqrt{2}\, n\sigma)^{-1},</math>

and using <math>n = N/V = p/(k_\text{B}T)</math> ([[ideal gas law]]) and <math>\sigma =\pi d^2</math> (effective cross-sectional area for spherical particles with diameter <math>d</math>), it may be shown that the mean free path is<ref>{{cite web|url=http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/menfre.html |title=Mean Free Path, Molecular Collisions |publisher=Hyperphysics.phy-astr.gsu.edu |access-date=2011-11-08}}</ref>

:<math>\ell = \frac{k_\text{B}T}{\sqrt 2 \pi d^2 p},</math>

where ''k''{{sub|B}} is the [[Boltzmann constant]], <math>p</math> is the pressure of the gas and <math>T</math> is the absolute temperature.

In practice, the diameter of gas molecules is not well defined. In fact, the [[kinetic diameter]] of a molecule is defined in terms of the mean free path. Typically, gas molecules do not behave like hard spheres, but rather attract each other at larger distances and repel each other at shorter distances, as can be described with a [[Lennard-Jones potential]]. One way to deal with such "soft" molecules is to use the Lennard-Jones σ parameter as the diameter. 

Another way is to assume a hard-sphere gas that has the same [[dynamic viscosity|viscosity]] as the actual gas being considered.  This leads to a mean free path <ref>{{cite book|title=Introduction to physical gas dynamics|year=1965|publisher=Krieger Publishing Company|author=Vincenti, W. G. and Kruger, C. H.|page=414}}</ref>

:<math>\ell = \frac{\mu}{\rho} \sqrt{\frac{\pi m}{2 k_\text{B}T}}=\frac{\mu}{p} \sqrt{\frac{\pi k_\text{B}T}{2 m}},</math>

where <math>m </math> is the [[molecular mass]], <math>\rho= m p/(k_\text{B}T)</math> is the density of ideal gas, and ''&mu;'' is the dynamic viscosity. This expression can be put into the following convenient form

:<math>\ell = \frac{\mu}{p} \sqrt{\frac{\pi R_{\rm specific}T}{2}},</math>

with <math> R_{\rm specific}=k_\text{B}/m </math> being the [[specific gas constant]], equal to 287 J/(kg*K) for air. Viscosity ''&mu;'' is low, 18.5 μPa·s at (25 °C, 1 bar), and p-dependent.  

The following table lists some typical values for air at different pressures at room temperature. Note that different definitions of the molecular diameter, as well as different assumptions about the value of atmospheric pressure (100 vs 101.3 kPa) and room temperature (293.15 vs 296.15 K (20-23 °C) or even 300 K) can lead to slightly different values of the mean free path.
{| class="wikitable"
|-
! style="width:16%;"|Vacuum range
! style="width:16%;"|[[Pressure]] in [[pascal (unit)|hPa]] ([[Bar (unit)|mbar]])
! style="width:16%;"|Pressure in [[mmHg]] ([[Torr]])
! style="width:16%;"|[[number density]] ([[Molecules]] / cm<sup>3</sup>)
! style="width:16%;"|number density (Molecules / m<sup>3</sup>)
! style="width:16%;"|Mean free path
|-
| Ambient pressure
| 1013
| 759.8
| 2.7 × 10<sup>19</sup>
| 2.7 × 10<sup>25</sup> <ref><math display="inline">n_0=p_{atm}/(k_BT_{std})</math> [[Loschmidt constant]]</ref>
| 64 – 68 [[Nanometre|nm]]<ref>{{cite journal|last1=Jennings|first1=S|title=The mean free path in air|journal=Journal of Aerosol Science|volume=19|page=159|year=1988|doi=10.1016/0021-8502(88)90219-4|issue=2|bibcode=1988JAerS..19..159J}}</ref>
|-
| Low vacuum
| 300 – 1
| 220 – 8×10<sup>−1</sup>
|10<sup>19</sup> – 10<sup>16</sup>
| 10<sup>25</sup> – 10<sup>22</sup>
| 0.1 – 100 [[Micrometre|μm]]
|-
| Medium vacuum
| 1 – 10<sup>−3</sup> (0.1 Pa)
| 8×10<sup>−1</sup> – 8×10<sup>−4</sup>
| 10<sup>16</sup> – 10<sup>13</sup>
| 10<sup>22</sup> – 10<sup>19</sup>
| 0.1 – 100&nbsp;mm
|-
| High vacuum
| 10<sup>−3</sup> – 10<sup>−7</sup> (10 μPa)
| 8×10<sup>−4</sup> – 8×10<sup>−8</sup>
| 10<sup>13</sup> – 10<sup>9</sup>
| 10<sup>19</sup> – 10<sup>15</sup>
| 10&nbsp;cm – 1&nbsp;km
|-
| Ultra-high vacuum
| 10<sup>−7</sup> – 10<sup>−12</sup> (0.1 nPa)
| 8×10<sup>−8</sup> – 8×10<sup>−13</sup>
| 10<sup>9</sup> – 10<sup>4</sup>
| 10<sup>15</sup> – 10<sup>10</sup>
| 1&nbsp;km – 10<sup>5</sup> km
|-
| Extremely high vacuum
| <10<sup>−12</sup>
| <8×10<sup>−13</sup>
| <10<sup>4</sup>
| <10<sup>10</sup>
| >10<sup>5</sup> km
|}