Section: 6 | Mean Free Path and Related Properties of Gases |
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In the simplest version of the kinetic theory of gases, molecules are treated as hard spheres of diameter *d* which make binary collisions only. In this approximation the mean distance traveled by a molecule between successive collisions, the mean free path* l*, is related to the collision diameter by:

$l=\frac{kT}{\pi \sqrt{2}P{d}^{2}}$

where *P* is the pressure, *T* the absolute temperature, and *k* the Boltzmann constant. At standard conditions (*P* = 100,000 Pa and *T* = 298.15 K), this relation becomes:

$l=\frac{9.27\cdot {10}^{27}}{{d}^{2}}$

where *l* and *d *are in meters.

Using the same model and the same standard pressure, the collision diameter can be calculated from the viscosity *η* by the kinetic theory relation:

$\eta =\frac{2.67\cdot {10}^{-20}{(MT)}^{1/2}}{{d}^{2}}$

where *η* is in units of µPa s and *M* is the molar mass in g mol^{-1}. Kinetic theory also gives a relation for the mean velocity *v̄* of molecules of mass* m*:

$\overline{v}={\left(\frac{8kT}{\pi m}\right)}^{1/2}=145.5{(T\text{}/\text{}M)}^{1/2}\text{m}\text{}\text{/}\text{}\text{s}$

Finally, the mean time *τ* between collisions can be calculated from the relation *τv̄ *= *l*, or *τ* = *l*/*v̄*; for argon, *τ* = 72.3 nm / 397 m s^{-1} = 0.182 × 10^{-9} s = 182 ps.

The table below gives values of the collision diameter, mean free path, mean velocity, and mean time betweeen collisons for some common gases at 25 °C and atmospheric pressure*,* all calculated from measured gas viscosities (see Refs. 2 and 3 and the table “Viscosity of Gases” in this section). Column definitions are as follows.

Column heading |
Definition |

Name |
Name of gas |

Mol. form. |
Molecular formula of gas |

d |
Collision diameter, in nm |

l |
Mean free path, in nm |

v̄ |
Mean velocity, in m s^{-1} |

τ |
Mean time between collisions, in ps |

It is seen from the above equations that the mean free path varies directly with *T* and inversely with *P, *while the mean velocity varies as the square root of *T* and, in this approximation, is independent of *P*.

A more accurate model, in which molecular interactions are described by a Lennard-Jones potential, gives mean free path values about 5% lower than this table (see Ref. 4).

- Reid, R. C., Prausnitz, J. M., and Poling, B. E.,
*The Properties of Gases and Liquids, Fourth Edition*, McGraw-Hill, New York, 1987. - Lide, D. R., and Kehiaian, H. V.,
*CRC Handbook of Thermophysical and Thermochemical Data*, CRC Press, Boca Raton, FL, 1994. - Vargaftik, N. B.,
*Tables of Thermophysical Properties of Liquids and Gases, Second Edition*, John Wiley, New York, 1975. [https://doi.org/10.1007/978-3-642-52504-9_13] - Kaye, G. W. C., and Laby, T. H.,
*Tables of Physical and Chemical Constants, 15th Edition,*Longman, London, 1986.

Name | Mol. form. | d/nm | l/nm | v̄/m s^{–1} | τ/ps |

Air | 0.366 | 69.1 | 467 | 148 | |

Ammonia | NH_{3} | 0.432 | 49.9 | 609 | 82 |

Argon | Ar | 0.358 | 72.3 | 397 | 182 |

Carbon dioxide | CO_{2} | 0.453 | 45.1 | 379 | 119 |

Helium | He | 0.215 | 200 | 1256 | 159 |

Hydrogen | H_{2} | 0.271 | 126 | 1769 | 71 |

Krypton | Kr | 0.408 | 55.6 | 274 | 203 |

Neon | Ne | 0.254 | 143 | 559 | 256 |

Nitrogen | N_{2} | 0.370 | 67.5 | 475 | 142 |

Oxygen | O_{2} | 0.355 | 73.7 | 444 | 166 |

Xenon | Xe | 0.478 | 40.5 | 219 | 185 |