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 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]

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

List of diameters

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

Molecule		Molecular weight	Kinetic diameter (pm)	ref
Name	Formula	Molecular weight	Kinetic diameter (pm)	ref
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,

\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,

\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.[7]

References

Joos & Freeman, p. 573
Ismail et al., p. 14
Freude, p. 4
Matteucci et al., p. 6
Breck
Li & Talu, p. 373
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.

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[1] Joos & Freeman, p. 573

[Ismail14-2] Ismail et al., p. 14

[3] Freude, p. 4

[Matteucci6-4] Matteucci et al., p. 6

[Breck-5] Breck

[6] Li & Talu, p. 373

[7] Freude, pp. 3-4