Section: 6 | Mean Free Path and Related Properties of Gases |
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 The recommended form of citation is: John R. Rumble, ed., CRC Handbook of Chemistry and Physics, 103rd Edition (Internet Version 2022), CRC Press/Taylor & Francis, Boca Raton, FL. If a specific table is cited, use the format: "Physical Constants of Organic Compounds," in CRC Handbook of Chemistry and Physics, 103rd Edition (Internet Version 2022), John R. Rumble, ed., CRC Press/Taylor & Francis, Boca Raton, FL.

# MEAN FREE PATH AND RELATED PROPERTIES OF GASES

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:

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

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 of molecules of mass m:

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

Finally, the mean time τ between collisions can be calculated from the relation τv̄ = l, or τ = l/;  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 for the table 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).

# References

1. Reid, R. C., Prausnitz, J. M., and Poling, B. E., The Properties of Gases and Liquids, Fourth Edition, McGraw-Hill, New York, 1987.
2. Lide, D. R., and Kehiaian, H. V., CRC Handbook of Thermophysical and Thermochemical Data, CRC Press, Boca Raton, FL, 1994.
3. 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]
4. Kaye, G. W. C., and Laby, T. H., Tables of Physical and Chemical Constants, 15th Edition, Longman, London, 1986.

## Mean Free Path, Mean Velocity, and Time between Collisions for Common Gases

 Name Mol. form. d/nm l/nm v̄/m s–1 τ/ps Air 0.366 69.1 467 148 Ammonia NH3 0.432 49.9 609 82 Argon Ar 0.358 72.3 397 182 Carbon dioxide CO2 0.453 45.1 379 119 Helium He 0.215 200 1256 159 Hydrogen H2 0.271 126 1769 71 Krypton Kr 0.408 55.6 274 203 Neon Ne 0.254 143 559 256 Nitrogen N2 0.370 67.5 475 142 Oxygen O2 0.355 73.7 444 166 Xenon Xe 0.478 40.5 219 185

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