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]

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

which has units of [volume][time]⁻¹.

Here,

• ${\displaystyle N_{\text{A}}}$ is the number of A molecules in the gas,
• ${\displaystyle N_{\text{B}}}$ is the number of B molecules in the gas,
• ${\displaystyle \sigma _{\text{AB}}}$ is the collision cross section, the "effective area" seen by two colliding molecules, simplified to ${\displaystyle \sigma _{\text{AB}}=\pi (r_{\text{A}}+r_{\text{B}})^{2}}$, where ${\displaystyle r_{\text{A}}}$ the radius of A and ${\displaystyle r_{\text{B}}}$ the radius of B.
• ${\displaystyle k_{\text{B}}}$ is the Boltzmann constant,
• ${\displaystyle T}$ is the temperature,
• ${\displaystyle \mu _{\text{AB}}}$ is the reduced mass of the reactants A and B, ${\displaystyle \mu _{\text{AB}}={\frac {{m_{\text{A}}}{m_{\text{B}}}}{{m_{\text{A}}}+{m_{\text{B}}}}}}$

Collision in diluted solution

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 (unit K)
• ${\displaystyle \eta }$ is the viscosity of the solution (pascal seconds)
• ${\displaystyle n}$ is the concentration of particles per cm³

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