Section: 6 | Mean Free Path and Related Properties of Gases |

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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:

$l = \frac{k T}{π \sqrt{2} P d^{2}}$

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:

$η = \frac{2.67 \cdot 10^{- 20} {(M T)}^{1 / 2}}{d^{2}}$

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:

$\bar{v} = {(\frac{8 k T}{π m})}^{1 / 2} = 145.5 {(T / M)}^{1 / 2} m / s$

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 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

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.

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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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MEAN FREE PATH AND RELATED PROPERTIES OF GASES

References

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

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