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

1. chem.libretexts.org: Collision Frequency
2. Debye, P. (1942). "Reaction Rates in Ionic Solutions". Transactions of the Electrochemical Society. 82 (1): 265. doi:10.1149/1.3071413. ISSN 0096-4743.