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]