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:

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

which has units of [volume][time]⁻¹.

Here,

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

Collision in diluted solution

In the case of equal-size particles at a concentration $n$ 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 (unit K)
• $\eta$ is the viscosity of the solution (pascal seconds)
• $n$ 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 $\nu$.