# 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.

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,

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

$d^{2}={1 \over {\sqrt {2}}\pi ln}$ ## List of diameters

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

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

## 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.