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 is^[1] $${\displaystyle Z=N_{\text{A}}{N}_{\text{B}}{\sigma }_{\text{AB}}\sqrt{\frac{8k_{\text{B}}T}{\pi {\mu }_{\text{AB}}}},}$$ 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 as^[2] $${\displaystyle \nu =\frac{8k_{\text{B}}T}{3\eta }n,}$$ 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]

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