Chemistry:Collision frequency

Short description: Physics calculation for collisions

Main article: Chemistry: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 $\nu$.