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]

$d^2 = {1 \over \pi l n}$

where,

d is the kinetic diameter,

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]

$d^2 = {1 \over \sqrt 2 \pi l n}$

List of diameters

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

Molecule		Molecular weight	Kinetic diameter (pm)	ref
Name	Formula
Hydrogen	H2	2	289	[2]
Helium	He	4	260	[4]
Methane	CH4	16	380	[2]
Water	H2O	18	265	[2]
Nitrogen	N2	28	364	[2]
Carbon monoxide	CO	28	376	[4]
Ethylene	C2H4	28	390	[4]
Oxygen	O2	32	346	[2]
Hydrogen sulfide	H2S	34	360	[4]
Propylene	C3H6	42	450	[4]
Carbon dioxide	CO2	44	330	[2]
Propane	C3H8	44	430	[4]
Benzene	C6H6	78	585	[5]

Dissimilar particles

The colliding particle may be a different molecule to the molecules in the gas and hence will have a different kinetic diameter. This can occur, for instance, if a beam of fast particles is fired into the gas. The kinetic diameter defines an area, σ, called the scattering cross section. The scattering cross section in a collision is defined by the sum of the kinetic diameters of the two particles,

$\sigma = \pi (r_1 + r_2)^2$

where.

r₁, r₂ are, respectively half the kinetic diameters, ie radii, of the two particles.

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

$\alpha \equiv n \sigma$

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

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

For similar particles, r₁ = r₂ and,

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

as before.^[6]

References

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

Bibliography

• Freude, D., Molecular Physics, chapter 2, 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.