# Kinetic diameter

Kinetic diameter is a measure applied to atoms and molecules that expresses the likelihood that a molecule in a gas will collide with another molecule. It is an indication of the size of the molecule as a target. The kinetic diameter is not the same as atomic diameter defined in terms of the size of the atom's electron shell, which is generally a lot smaller, depending on the exact definition used. Rather, it is the size of the sphere of influence that can lead to a scattering event.[1]

Kinetic diameter is related to the mean free path of molecules in a gas. Mean free path is the average distance that a particle will travel without collision. For a fast moving particle (that is, one moving much faster than the particles it is moving through) the kinetic diameter is given by,[2]

${\displaystyle d^{2}={1 \over \pi ln}}$
where,
d is the kinetic diameter,
r is the kinetic radius, r = d/2,
l is the mean free path, and
n is the number density of particles

However, a more usual situation is that the colliding particle being considered is indistinguishable from the population of particles in general. Here, the Maxwell–Boltzmann distribution of energies must be considered, which leads to the modified expression,[3]

${\displaystyle d^{2}={1 \over {\sqrt {2}}\pi ln}}$

## List of diameters

The following table lists the kinetic diameters of some common molecules;

Molecule Molecular
mass
Kinetic
diameter
(pm)
ref
Name Formula
Hydrogen H2 2 289 [2]
Helium He 4 260 [4]
Methane CH4 16 380 [2]
Ammonia NH3 17 260 [5]
Water H2O 18 265 [2]
Neon Ne 20 275 [5]
Acetylene C2H2 26 330 [5]
Nitrogen N2 28 364 [2]
Carbon monoxide CO 28 376 [4]
Ethylene C2H4 28 390 [4]
Nitric oxide NO 30 317 [4]
Oxygen O2 32 346 [2]
Hydrogen sulfide H2S 34 360 [4]
Hydrogen chloride HCl 36 320 [5]
Argon Ar 40 340 [5]
Propylene C3H6 42 450 [4]
Carbon dioxide CO2 44 330 [2]
Nitrous oxide N2O 44 330 [4]
Propane C3H8 44 430 [4]
Sulfur dioxide SO2 64 360 [5]
Chlorine Cl2 70 320 [5]
Benzene C6H6 78 585 [6]
Hydrogen bromide HBr 81 350 [5]
Krypton Kr 84 360 [5]
Xenon Xe 131 396 [5]
Sulfur hexafluoride SF6 146 550 [5]
Carbon tetrachloride CCl4 154 590 [5]
Bromine Br2 160 350 [5]

## Dissimilar particles

Collisions between two dissimilar particles occur when a beam of fast particles is fired into a gas consisting of another type of particle, or two dissimilar molecules randomly collide in a gas mixture. For such cases, the above formula for scattering cross section has to be modified.

The scattering cross section, σ, in a collision between two dissimilar particles or molecules is defined by the sum of the kinetic diameters of the two particles,

${\displaystyle \sigma =\pi (r_{1}+r_{2})^{2}}$
where.
r1, r2 are, half the kinetic diameter (ie, the kinetic radii) of the two particles, respectively.

We define an intensive quantity, the scattering coefficient α, as the product of the gas number density and the scattering cross section,

${\displaystyle \alpha \equiv n\sigma }$

The mean free path is the inverse of the scattering coefficient,

${\displaystyle l={1 \over \alpha }={1 \over \sigma n}}$

For similar particles, r1 = r2 and,

${\displaystyle l={1 \over \sigma n}={1 \over 4\pi r^{2}n}={1 \over \pi d^{2}n}}$

as before.[7]

## References

1. ^ Joos & Freeman, p. 573
2. Ismail et al., p. 14
3. ^ Freude, p. 4
4. Matteucci et al., p. 6
5. Breck
6. ^ Li & Talu, p. 373
7. ^ Freude, pp. 3-4

## Bibliography

• Breck, Donald W., "Zeolite Molecular Sieves: Structure, Chemistry, and Use", New York: Wiley, 1974 ISBN 0471099856.
• Freude, D., Molecular Physics, chapter 2, 2004 unpublished draft, retrieved and archived 18 October 2015.
• Ismail, Ahmad Fauzi; Khulbe, Kailash; Matsuura, Takeshi, Gas Separation Membranes: Polymeric and Inorganic, Springer, 2015 ISBN 3319010956.
• Joos, Georg; Freeman, Ira Maximilian, Theoretical Physics, Courier Corporation, 1958 ISBN 0486652270.
• Li, Jian-Min; Talu, Orhan, "Effect of structural heterogeneity on multicomponent adsorption: benzene and p-xylene mixture on silicalite", in Suzuki, Motoyuki (ed), Fundamentals of Adsorption, pp. 373-380, Elsevier, 1993 ISBN 0080887724.
• Matteucci, Scott; Yampolskii, Yuri; Freeman, Benny D.; Pinnau, Ingo, "Transport of gases and vapors in glassy and rubbery polymers" in, Yampolskii, Yuri; Freeman, Benny D.; Pinnau, Ingo, Materials Science of Membranes for Gas and Vapor Separation, pp. 1-47, John Wiley & Sons, 2006 ISBN 0470029048.