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 entities of species A and species B is¹ $${\displaystyle Z=N_{\text{A}}N_{\text{B}}\sigma _{\text{AB}}{\sqrt {\frac {8k_{\text{B}}T}{\pi \mu _{\text{AB}}}}},}$$ where

${\displaystyle N_{\text{A}}}$ is the number of A particles in the volume,

${\displaystyle N_{\text{B}}}$ is the number of B particles in the volume,

${\displaystyle \sigma _{\text{AB}}}$ is the collision cross section, the "effective area" seen by two colliding molecules (for hard spheres, ${\displaystyle \sigma _{\text{AB}}=\pi (r_{\text{A}}+r_{\text{B}})^{2}}$, where ${\displaystyle r_{\text{A}}}$ is the radius of A, and ${\displaystyle r_{\text{B}}}$ is the radius of B),

${\displaystyle k_{\text{B}}}$ is the Boltzmann constant,

${\displaystyle T}$ is the thermodynamic temperature,

${\displaystyle \mu _{\text{AB}}={\frac {m_{\text{A}}m_{\text{B}}}{m_{\text{A}}+m_{\text{B}}}}}$ is the reduced mass of A and B particles.

In the case of equal-size particles at a concentration ${\displaystyle n}$ in a solution of viscosity ${\displaystyle \eta }$, an expression for collision frequency ${\displaystyle Z=V\nu }$, where ${\displaystyle V}$ is the volume in question, and ${\displaystyle \nu }$ is the number of collisions per second, can be written as² $${\displaystyle \nu ={\frac {8k_{\text{B}}T}{3\eta }}n,}$$ where

${\displaystyle k_{B}}$ is the Boltzmann constant,

${\displaystyle T}$ is the absolute temperature,

${\displaystyle \eta }$ is the viscosity of the solution,

${\displaystyle n}$ is the number density.

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