Collision frequency

Collision frequency describes the rate of collisions between two atomic or molecular species in a given volume, per unit time. In an ideal gas, assuming that the species behave like hard spheres, the collision frequency between A and B is:^[1]

${\displaystyle Z=N_{\text{A}}N_{\text{B}}\sigma _{AB}{\sqrt {\frac {8k_{\text{B}}T}{\pi \mu _{AB}}}},}$

SI unit of Z is number of collision per time ${\displaystyle m^{3}s^{-1}}$.

where:

• ${\displaystyle N_{A}}$ is the number of A molecules in the gas,
• ${\displaystyle N_{B}}$ is the number of B molecules in the gas,
• ${\displaystyle \sigma _{AB}}$ is the collision cross section (unit ${\displaystyle m^{2}}$), the area when two molecules collide with each other, simplified to ${\displaystyle \sigma _{AB}=\pi (r_{A}+r_{B})^{2}}$, where ${\displaystyle r_{A}}$ the radius of A and ${\displaystyle r_{B}}$ the radius of B.
• ${\displaystyle k_{B}}$ is the Boltzmann's constant (unit ${\displaystyle m^{2}\ kg\ s^{-2}\ K^{-1})}$,
• ${\displaystyle T}$ is the temperature (unit ${\displaystyle K}$),
• ${\displaystyle \mu _{AB}}$ is the reduced mass of the reactants A and B, ${\displaystyle \mu _{AB}={\frac {{m_{A}}{m_{B}}}{{m_{A}}+{m_{B}}}}}$ (unit ${\displaystyle kg}$)

Collision in diluted solution[]

Collision in diluted gas or liquid solution is regulated by diffusion instead of direct collisions, which can be calculated from Fick's laws of diffusion.

In the case of equal-size particles at a concentration ${\displaystyle n}$ in a solution of viscosity ${\displaystyle \eta }$ , an expression for collision frequency ${\displaystyle \nu =ZV}$ where ${\displaystyle V}$ is the volume in question, and ${\displaystyle \nu }$ is the number of collisions per second, can be written as :^[2]

${\displaystyle \nu ={\frac {8k_{B}T}{3\eta }}n,}$

Here the frequency is independent of particle size, a result noted as counter-intuitive. For particles of different size, more elaborate expressions can be derived for estimating ${\displaystyle \nu }$.^[2]

References[]