Collision frequency

Main article: Collision theory

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

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

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

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

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

$T$ is the thermodynamic temperature,

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

Collision in diluted solution

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

$k_{B}$ is the Boltzmann constant,

$T$ is the absolute temperature,

$\eta$ is the viscosity of the solution,

$n$ 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 $\nu$.